id int64 | problem_number string | title string | statement string | background string | difficulty_level_id int64 | status string | proposed_by string | proposed_year int64 | category_id int64 | set_id int64 | view_count int64 | favorite_count int64 | created_at timestamp[s] | updated_at timestamp[s] | published bool | category dict | difficulty dict | set dict | duplicates list | status_review unknown | archive_reason string | duplicate_of list | old_problem_number string | notes string | tags list | literature_assessment string | literature_sources list | literature_checked_at timestamp[s] | solution_reference string | source_url string | source string | prize string | references list | clean_statement string | original_statement string | statement_status string | statement_verification string | research_classification string | research_summary string | research_difficulty_suggested string | historical_research_summary string | research_summary_reviewed_at timestamp[s] | source_citation string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
116 | GREEN-028 | Gowers Box Norms over Finite Fields | Let $p$ be an odd prime and suppose $f : \mathbb{F}_p^n \times \mathbb{F}_p^n \to \mathbb{C}$ is bounded pointwise by 1. Suppose $\mathbb{E}_h \|\Delta_{(h,h)}f\|_\square^4 \geq \delta$. Does $f$ correlate with a function of the form $a(x)b(y)c(x+y)(-1)^{q(x,y)}$? | This asks for an inverse theorem for a particular Gowers norm in product spaces over finite fields. Understanding which structured functions correlate with high Gowers norm is central to higher-order Fourier analysis.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Clas... | 2 | open | null | null | 2 | 3 | 84 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
117 | GREEN-029 | Inverse Theorem for Gowers Norms | Determine bounds for the inverse theorem for Gowers norms. | The inverse theorem characterizes functions with large Gowers $U^{s+1}$ norm. Leng-Sah-Sawhney (2024) established a quasi-polynomial inverse theorem for $\|\cdot\|_{U^{s+1}[N]}$ norms for all $s \geq 3$. Improving to polynomial bounds remains a major challenge in additive combinatorics.
<!-- LITERATURE-TRIAGE:BEGIN --... | 2 | partially_solved | null | null | 2 | 3 | 95 | 6 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
118 | GREEN-030 | Φ(G) and Φ'(G) Coincidence | Do $\Phi(G)$ and $\Phi'(G)$ coincide? | This asks whether two different notions of the Frattini-like subgroup of $G$ are equal. The Frattini subgroup consists of non-generators; different definitions can arise in different contexts. Determining their equivalence has implications for group theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checke... | 1 | partially_solved | null | null | 4 | 3 | 73 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
119 | GREEN-031 | Sumsets Containing Composites | Suppose $A, B \subset \{1, \dots, N\}$ both have size $N^{0.49}$. Does $A + B$ contain a composite number? | This asks whether sumsets of moderately large sets must contain composite numbers. Since primes have density $1/\log N$, sets of size $N^{0.49}$ are much denser, suggesting their sumset should hit composites. However, proving this rigorously requires understanding the additive structure of primes.
<!-- LITERATURE-TRIA... | 1 | open | null | null | 1 | 3 | 81 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
120 | GREEN-032 | Sums of Smooth Numbers | Is every $n \leq N$ the sum of two integers, all of whose prime factors are at most $N^\varepsilon$? | Smooth numbers have only small prime factors. This asks if every number is a sum of two smooth numbers, which would show smooth numbers have excellent additive properties. Such a result would have implications for number theory and the distribution of smooth numbers.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature revi... | 2 | partially_solved | null | null | 1 | 3 | 88 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
121 | GREEN-033 | Sumsets of Perfect Squares | Is there an absolute constant $c > 0$ such that if $A \subset \mathbb{N}$ is a set of squares of size at least 2, then $|A + A| \geq |A|^{1+c}$? | This asks whether sets of perfect squares have superlinear sumset growth. Squares are highly structured (sparse in $\mathbb{N}$), and one expects their sumsets to grow substantially. Determining the optimal exponent $c$ is a fundamental problem in additive number theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature ... | 2 | partially_solved | null | null | 1 | 3 | 92 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
122 | GREEN-034 | Covering Squares with Sumsets | Suppose $A + A$ contains the first $n$ squares. Is $|A| \geq n^{1-o(1)}$? | If a set's sumset contains all squares up to $n^2$, must the set have size nearly $n$? This explores the inverse problem: given that a sumset covers a structured set (squares), what can we say about the original set?
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solv... | 1 | partially_solved | null | null | 1 | 3 | 85 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
123 | GREEN-035 | Products of Primes Modulo p | Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \in \{1, \dots, p-1\}$ congruent to some product $a_1a_2$ modulo $p$? | This asks whether pairwise products of primes cover all residues modulo $p$. Matom\"aki-Ter\"av\"ainen (2023) made significant progress, showing that products of three primes suffice. Whether two primes suffice remains open.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partia... | 2 | partially_solved | null | null | 1 | 3 | 96 | 6 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
124 | GREEN-036 | Multiplicatively Closed Set Density | Let $A$ be the smallest set containing 2 and 3, and closed under the operation $a_1a_2 - 1$ (if $a_1, a_2 \in A$, then $a_1a_2 - 1 \in A$). Does $A$ have positive density? | This defines a set generated by a multiplicative-like operation. Understanding its density is nontrivial because the operation $a_1a_2 - 1$ mixes multiplication with additive structure. Whether such sets have positive density connects number theory with dynamical systems.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature... | 2 | open | null | null | 1 | 3 | 77 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
125 | GREEN-037 | Primes with p-2 Having Odd Omega | Do there exist infinitely many primes $p$ for which $p-2$ has an odd number of prime factors (counting multiplicity)? | This asks about the parity of $\Omega(p-2)$ where $\Omega(n)$ counts prime factors with multiplicity. Since $p-2$ is even for odd primes $p > 2$, we're asking about the structure of $(p-2)/2$. This is a prime-shifted multiplicative function question.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-... | 1 | open | null | null | 1 | 3 | 83 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
126 | GREEN-038 | Difference Sets Containing Squares | Is there $c > 0$ such that whenever $A \subset [N]$ has size $N^{1-c}$, the difference set $A - A$ contains a nonzero square? | This asks how large a set must be to guarantee its difference set contains a square. Similarly one can ask if $A - A$ contains a prime minus one. These questions probe the additive structure forced by density.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**... | 1 | partially_solved | null | null | 1 | 3 | 89 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
127 | GREEN-039 | Gaps Between Sums of Two Squares | Is there always a sum of two squares between $X - \frac{1}{10}X^{1/4}$ and $X$? | Sums of two squares have density $c/\sqrt{\log X}$, so gaps can be large. This asks for an upper bound on the largest gap. Such results would improve our understanding of the distribution of representable numbers in quadratic forms.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:*... | 2 | partially_solved | null | null | 1 | 3 | 91 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
128 | GREEN-040 | Waring's Problem Over Finite Fields | Determine bounds for Waring's problem over finite fields. | Waring's problem asks: can every element be written as a sum of $k$ $d$-th powers? Over finite fields $\mathbb{F}_q$, the problem has different character. Determining the minimum $k$ for given $d$ and $q$ is a classical problem in algebraic number theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked ... | 2 | partially_solved | null | null | 1 | 3 | 86 | 5 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "scope_correction",
"evidence": "scope_review"
} | null | null | null | null | null | Sawin obtains an asymptotic formula for Waring’s problem over function fields, resolving Green’s original Problem 67 in its stated range. The imported record instead asks for unspecified bounds over finite fields and describes a different problem; the source theorem cannot certify this wording as solved.
**What remain... | [
{
"url": "https://arxiv.org/abs/2412.14053",
"label": "Will Sawin, The asymptotic in Waring’s problem over function fields via a singular locus in the circle method (2024)"
},
{
"url": "https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf",
"label": "Ben Green, 100 Open Problems, Probl... | 2026-09-27T00:00:00 | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
129 | GREEN-041 | Cubic Curves in F_p^2 | Suppose $A \subset \mathbb{F}_p^2$ is a set meeting every line in at most 2 points. Is it true that all except $o(p)$ points of $A$ lie on a cubic curve? | Sets avoiding three collinear points have special structure. Over finite fields, the Hasse-Weil bound and algebraic geometry suggest such sets should lie nearly on a cubic. This is a finite-field analogue of classical incidence geometry.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Sta... | 2 | partially_solved | null | null | 6 | 3 | 84 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
130 | GREEN-042 | Collinear Triples and Cubic Curves | Fix $k$. Let $A \subset \mathbb{R}^2$ be a set of $n$ points with no more than $k$ on any line. Suppose at least $\delta n^2$ pairs $(x, y) \in A \times A$ have the line $xy$ containing a third point of $A$. Is there a cubic curve containing at least $cn$ points of $A$? | If many pairs determine lines through a third point, the set should have algebraic structure. This asks if a cubic curve captures this structure. It generalizes results from the joints problem and incidence geometry.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solv... | 2 | partially_solved | null | null | 6 | 3 | 78 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
131 | GREEN-043 | Erdős-Szekeres with Visibility | Fix integers $k, \ell$. Given $n \geq n_0(k, \ell)$ points in $\mathbb{R}^2$, is there either a line containing $k$ of them, or $\ell$ of them that are mutually visible? | This is a Ramsey-type problem mixing collinearity and visibility (no point blocks the segment between two others). It generalizes the Erdős-Szekeres theorem to a geometric context, asking for unavoidable configurations.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_s... | 1 | partially_solved | null | null | 6 | 3 | 81 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
132 | GREEN-044 | Collinear 4-tuples Force Collinear 5-tuples | Suppose $A \subset \mathbb{R}^2$ is a set of size $n$ with $cn^2$ collinear 4-tuples. Does it contain 5 points on a line? | Many collinear 4-tuples suggest the set has strong linear structure. This asks if this forces an actual line through 5 points. It's related to the Szemerédi-Trotter theorem and incidence bounds.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:... | 1 | partially_solved | null | null | 6 | 3 | 75 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
133 | GREEN-045 | No Three in Line in [N]^2 | What is the largest subset of the grid $[N]^2$ with no three points on a line? In particular, for $N$ sufficiently large, is it impossible to have a set of size $2N$ with this property? | The cap set problem in two dimensions. Erdős conjectured sets of size $O(N)$ exist, but proving or disproving this remains open. The problem connects discrete geometry, additive combinatorics, and the polynomial method.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_s... | 2 | partially_solved | null | null | 6 | 3 | 94 | 6 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
134 | GREEN-046 | Smooth Surfaces Intersecting 2-planes | Let $\Gamma$ be a smooth codimension 2 surface in $\mathbb{R}^n$. Must $\Gamma$ intersect some 2-dimensional plane in 5 points, if $n$ is sufficiently large? | This asks about unavoidable intersection patterns between smooth surfaces and planes in high dimensions. It's related to the Kakeya problem and incidence geometry in higher dimensions.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL... | 2 | partially_solved | null | null | 6 | 3 | 71 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
135 | GREEN-047 | No 5 Points on 2-plane in [N]^d | What is the largest subset of $[N]^d$ with no 5 points on a 2-plane? | This generalizes the no-three-in-line problem to higher dimensions and 2-planes. Determining the maximum size of such sets involves combinatorial geometry and higher-dimensional incidence bounds.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRI... | 1 | open | null | null | 6 | 3 | 76 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
136 | GREEN-048 | Balanced Ham Sandwich Line | Let $X \subset \mathbb{R}^2$ be a set of $n$ points. Does there exist a line $\ell$ through at least two points of $X$ such that the numbers of points on either side of $\ell$ differ by at most 100? | This is a variant of the ham sandwich theorem asking for a balanced bisector that passes through points of the set. The classical ham sandwich theorem doesn't require passing through points, making this version more constrained.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** op... | 1 | open | null | null | 6 | 3 | 79 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
137 | GREEN-049 | Sparse Hitting Set for Rectangles | Let $A$ be a set of $n$ points in the plane. Can one select $A' \subset A$ of size $n/2$ such that any axis-parallel rectangle containing 1000 points of $A$ contains at least one point of $A'$? | This asks for an efficient hitting set for rectangles defined by a point set. Such results have applications in computational geometry and range searching.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** ... | 1 | open | null | null | 6 | 3 | 74 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
138 | GREEN-050 | Small Triangles in the Unit Disc | Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them? | This asks about unavoidable small-area triangles in dense point sets. Cohen-Pohoata-Zakharov (2023) improved the bound to $n^{-8/7-c}$. Reaching the conjectured $n^{-2}$ bound remains open and connects to the Heilbronn triangle problem.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Stat... | 2 | partially_solved | null | null | 6 | 3 | 88 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
139 | GREEN-051 | Axis-Parallel Rectangles in Dense Sets | Suppose $A$ is an open subset of $[0, 1]^2$ with measure $\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\geq c\alpha^2$? | This asks if dense sets in the unit square must contain large axis-parallel rectangles. It's a continuous analogue of combinatorial rectangle problems and relates to measure-theoretic ergodic theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN... | 1 | open | null | null | 6 | 3 | 72 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
140 | GREEN-052 | Equidistribution of Integer Multiples | Let $c > 0$ and let $A$ be a set of $n$ distinct integers. Does there exist $\theta$ such that no interval of length $\frac{1}{n}$ in $\mathbb{R}/\mathbb{Z}$ contains more than $n^c$ of the numbers $\theta a \pmod 1$, for $a \in A$? | This asks about finding angles $\theta$ that spread out the set $A$ modulo 1. It's related to discrepancy theory and the distribution of sequences modulo 1.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literat... | 1 | partially_solved | null | null | 1 | 3 | 68 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
141 | GREEN-053 | Random Permutations Fixing k-Sets | Let $p(k)$ be the limit as $n \to \infty$ of the probability that a random permutation on $[n]$ preserves some set of size $k$. Is $p(k)$ a decreasing function of $k$? Is $p(k) = (C + o(1))k^{-\alpha}(\log k)^{-3/2}$ for some absolute constant $C$? | This concerns the probability that a random permutation fixes some subset. Eberhard-Ford-Green established asymptotic formulas. The question asks for monotonicity and precise asymptotics, connecting combinatorics and probability.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** p... | 2 | partially_solved | null | null | 2 | 3 | 75 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
142 | GREEN-054 | Comparable Elements in Integer Lattices | Consider a set $S \subset [N]^3$ with the property that any two distinct elements $s, s'$ of $S$ are comparable (in the coordinatewise partial order). Is $|S| \leq N^{2-\delta}$ for some $\delta > 0$? | An antichain in $[N]^d$ can have size $\binom{N}{d/2}^d \sim N^{d/2}$. This asks if totally comparable sets (chains) in 3D are even smaller, achieving $N^{2-\delta}$ rather than $N^2$. It's a question in extremal combinatorics and poset theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)... | 1 | open | null | null | 2 | 3 | 71 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
143 | GREEN-055 | Stable Density on Subspaces | Let $A \subset \mathbb{F}_2^n$. If $V$ is a subspace, write $\alpha(V)$ for the density of $A$ on $V$. Is there some $V$ of moderately small codimension on which $\alpha$ is stable? | This asks if every set has a subspace where its density doesn't fluctuate wildly. Stability of density on subspaces is fundamental in additive combinatorics over $\mathbb{F}_2^n$ and relates to the polynomial Freiman-Ruzsa conjecture.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status... | 2 | partially_solved | null | null | 2 | 3 | 77 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
144 | GREEN-056 | Almost Invariant Sets Under Affine Maps | Suppose $A \subset \mathbb{Z}/p\mathbb{Z}$ has density $\frac{1}{2}$. Under what conditions on $K$ can $A$ be almost invariant under all maps $\phi(x) = ax + b$ with $|a|, |b| \leq K$? | This asks when a set is nearly preserved under small affine transformations. Understanding which sets have this property connects to additive combinatorics, group actions, and the structure of dense sets in cyclic groups.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially... | 2 | partially_solved | null | null | 2 | 3 | 69 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
145 | GREEN-057 | Trace Reconstruction | Given a string $x \in \{0, 1\}^n$, let $\tilde{x}$ be obtained by deleting bits independently at random with probability $\frac{1}{2}$. How many independent traces $\tilde{x}_1, \dots, \tilde{x}_m$ are needed to reconstruct $x$ with probability 0.9? | This fundamental problem in computational complexity asks how many noisy observations suffice to recover the original string. Chase (2020) improved bounds to $e^{n^{1/5}\log^C n}$. Determining the optimal bound is a major open question.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Stat... | 2 | partially_solved | null | null | 2 | 3 | 82 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
146 | GREEN-058 | Irreducibility of Random {0,1} Polynomials | Is a random polynomial with coefficients in $\{0, 1\}$ and nonzero constant term almost surely irreducible? | This asks whether most polynomials with binary coefficients are irreducible over $\mathbb{Q}$. Bary-Soroker-Koukoulopoulos-Kozma (2023) made further progress. The problem connects number theory, probability, and algebraic geometry.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:**... | 2 | partially_solved | null | null | 1 | 3 | 76 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
149 | GREEN-061 | N Queens Problem Asymptotics | In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \times n$ chessboard? | The n-queens problem asks for the number of ways to place $n$ queens on an $n \times n$ board so none attack each other. Determining the asymptotic growth rate of $Q(n)$ is a famous open problem in combinatorics. Rough bounds are known but the exact constant remains elusive.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literat... | 1 | partially_solved | null | null | 2 | 3 | 94 | 6 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | [
"GREEN-097"
] | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | null | null | Reconciled with the existing literature review of 2026-08-17: Simkin established the requested asymptotic enumeration: Q(n)=((1±o(1)) n e^(-alpha))^n for alpha=1.942±0.003. This is a consistency correction based on that cited review; the discussion does not independently re-verify its proof claims.
**What remains.** T... | [
{
"url": "https://arxiv.org/abs/2107.13460",
"label": "M. Simkin, The number of n-queens configurations, arXiv:2107.13460 (2021)."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates and follow-ups"... | 2026-09-27T00:00:00 | M. Simkin, The number of n-queens configurations, arXiv:2107.13460 (2021).: https://arxiv.org/abs/2107.13460 | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
150 | GREEN-062 | Bounds for Birch's Theorem | Let $d \geq 3$ be odd. Give bounds on $\nu(d)$ such that if $n > \nu(d)$ then any homogeneous polynomial $F(\mathbf{x}) \in \mathbb{Z}[x_1, \dots, x_n]$ of degree $d$ has a nontrivial integer zero. | Birch's theorem guarantees that homogeneous polynomials of odd degree have nontrivial zeros if there are enough variables. Determining the optimal $\nu(d)$ is a central problem in Diophantine equations and algebraic number theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** ... | 2 | partially_solved | null | null | 1 | 3 | 73 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | [
"GREEN-098"
] | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
151 | GREEN-063 | Solutions to Polynomial Equations in Dense Sets | Finding a single solution to $F(x_1, \dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of solutions in $A$ is roughly $\alpha^n$ times the number in $[X]$? | This asks when a dense set $A$ of density $\alpha$ contains the "expected" number of solutions to a Diophantine equation. Understanding when sparse sets behave like random sets for counting solutions is fundamental in analytic number theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
*... | 2 | partially_solved | null | null | 1 | 3 | 70 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | [
"GREEN-099"
] | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
152 | GREEN-064 | Residually Finite Groups | Is every group well-approximated by finite groups? | A group is residually finite if every nontrivial element has a nontrivial image in some finite quotient. This asks if all groups have this property. The answer is known to be no (infinite simple groups), but the question may refer to finitely generated/presented groups, where it remains interesting.
<!-- LITERATURE-TR... | 1 | partially_solved | null | null | 4 | 3 | 67 | 3 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | [
"GREEN-100"
] | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | null | null | Reconciled with the existing literature review of 2026-08-17: Under the standard interpretation that well-approximated by finite groups means residually finite, the universal assertion is false: a nontrivial infinite simple group has no nontrivial finite quotient. This is a consistency correction based on that cited re... | [
{
"url": "https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms/s1-26.1.61",
"label": "G. Higman, A Finitely Generated Infinite Simple Group, Journal of the London Mathematical Society 26 (1951), 61-64."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
... | 2026-09-27T00:00:00 | G. Higman, A Finitely Generated Infinite Simple Group, Journal of the London Mathematical Society 26 (1951), 61-64.: https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms/s1-26.1.61 | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
153 | GREEN-065 | Rado's Boundedness Conjecture | Suppose $a_1, \dots, a_k$ are integers which do not satisfy Rado's condition. Is $c(a_1, \dots, a_k)$ bounded in terms of $k$ only? | Rado's condition characterizes which linear equations are partition regular. For equations not satisfying this condition, $c(\cdot)$ is the minimum number of colors needed to avoid monochromatic solutions. Whether this depends only on $k$ (not the coefficients) is a fundamental question in Ramsey theory.
<!-- LITERATU... | 2 | partially_solved | null | null | 2 | 3 | 72 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
154 | GREEN-066 | Monochromatic x+y and xy | If $\{1, \dots, N\}$ is $r$-coloured, then for $N \geq N_0(r)$ there exist integers $x, y \geq 3$ such that $x+y$ and $xy$ have the same colour. Find reasonable bounds for $N_0(r)$. | This asks about unavoidable monochromatic additive-multiplicative patterns. Finding quantitative bounds for $N_0(r)$ connects Ramsey theory with both additive and multiplicative structure.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PAR... | 1 | partially_solved | null | null | 2 | 3 | 78 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
155 | GREEN-067 | Affine Translates of {0,1,3} | If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\{0, 1, 3\}$ that $A$ can contain? | This asks how many copies of the pattern $\{0, 1, 3\}$ (under affine transformations $x \mapsto ax + b$) can appear in an $n$-element set. Understanding maximal copies of specific patterns is fundamental in additive combinatorics.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** ... | 1 | open | null | null | 2 | 3 | 74 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
156 | GREEN-068 | Restricted Sumsets in Partitions | For which values of $k$ is the following true: whenever we partition $[N] = A_1 \cup \dots \cup A_k$, we have $|\bigcup_{i=1}^k (A_i \hat{+} A_i)| \geq \frac{1}{10} N$? | This asks how many parts are needed before restricted sumsets (sums of distinct elements) must cover a substantial fraction of $[N]$. The problem connects partition regularity with sumset structure.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-... | 2 | open | null | null | 2 | 3 | 68 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
157 | GREEN-069 | Sum of Cubes in F_3^n | Let $A_1, \dots, A_{100}$ be "cubes" in $\mathbb{F}_3^n$ (images of $\{0, 1\}^n$ under linear automorphisms). Is $A_1 + \dots + A_{100} = \mathbb{F}_3^n$? | This asks whether 100 cubes in $\mathbb{F}_3^n$ always sum to the entire space. It's a question about additive bases and the covering properties of structured sets in vector spaces over finite fields.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
**Classification:** SOL... | 2 | solved | null | null | 2 | 3 | 71 | 4 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "resolution",
"evidence": "published"
} | null | null | null | null | null | Yang Yu proves that four linear bases over F_3 already form an additive basis. Equivalently, four cubes suffice, which settles this record’s 100-cube assertion.
**What remains.** The analogous questions over other prime fields are separate and are not resolved here. | [
{
"url": "https://arxiv.org/abs/2510.01300",
"label": "Yang Yu, The Permanent Rank of a Matrix (Part Three) Note on the Additive Basis Conjecture (2025)"
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status ... | 2026-09-27T00:00:00 | Yang Yu, The Permanent Rank of a Matrix (Part Three) Note on the Additive Basis Conjecture (2025): https://arxiv.org/abs/2510.01300 | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
158 | GREEN-070 | Sets with No Unique Sum Representations | What is the size of the smallest set $A \subset \mathbb{Z}/p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation? | This asks for the minimum size of a set where every sum $a + a'$ has multiple representations. Bedert (2023) showed the answer lies between $\omega(p)\log p$ and $O(\log^2 p)$. Closing this gap would deepen our understanding of additive bases.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
... | 2 | partially_solved | null | null | 2 | 3 | 76 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
159 | GREEN-071 | Uniform Random Variables with Uniform Sum | Suppose $X, Y$ are finitely-supported independent random variables taking integer values such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly distributed on their ranges? | This asks if uniform sums force uniform summands. It's a discrete probability question with connections to additive combinatorics and the structure of convolutions.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current... | 1 | partially_solved | null | null | 2 | 3 | 70 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
160 | GREEN-072 | Large Subsets of Approximate Groups | Suppose $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \subset A$ with $|S| \gg K^{-O(1)}|A|$ and $S^8 \subset A^4$? | Approximate groups are sets with controlled doubling. This asks if they contain large subsets with even better multiplicative structure. Understanding approximate groups is central to geometric group theory and additive combinatorics.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status... | 2 | open | null | null | 4 | 3 | 69 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
161 | GREEN-073 | Structured Subsets with Bounded Doubling | Given a set $A \subset \mathbb{Z}$ with $D(A) \leq K$, find a large structured subset $A'$ which "obviously" has $D(A') \leq K + \varepsilon$. | Sets with small doubling constant have additive structure. This asks for an explicit, easily verifiable structured subset. Making structure "obvious" connects to algorithmic aspects of the Polynomial Freiman-Ruzsa conjecture.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** parti... | 2 | partially_solved | null | null | 2 | 3 | 68 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
162 | GREEN-074 | Sidon Set Size Bounds | Write $F(N)$ for the largest Sidon subset of $[N]$. Improve, at least for infinitely many $N$, the bounds $N^{1/2} + O(1) \leq F(N) \leq N^{1/2} + N^{1/4} + O(1)$. | Sidon sets have all pairwise sums distinct. The bounds have been tight for decades. Balogh-Füredi-Roy (2021) obtained a small improvement to the upper bound. Any further progress would be a major breakthrough in additive combinatorics.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Statu... | 2 | partially_solved | null | null | 2 | 3 | 89 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
163 | GREEN-075 | Large Gaps in Dilates | Let $p$ be a prime and let $A \subset \mathbb{Z}/p\mathbb{Z}$ be a set of size $\sqrt{p}$. Is there a dilate of $A$ with a gap of length $100\sqrt{p}$? | This asks whether dilates (multiplicative translates) of sets necessarily have large gaps. Understanding the distribution of dilates connects additive and multiplicative combinatorics in cyclic groups.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OP... | 1 | open | null | null | 2 | 3 | 72 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
164 | GREEN-076 | Optimal Sidon Bases | Are there infinitely many $q$ for which there is a set $A \subset \mathbb{Z}/q\mathbb{Z}$ with $|A| = (\sqrt{2} + o(1))q^{1/2}$ and $A + A = \mathbb{Z}/q\mathbb{Z}$? | This asks if Sidon-like sets (near-optimal density with few sum collisions) can form additive bases. The coefficient $\sqrt{2}$ is conjecturally optimal for such constructions.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current lite... | 2 | open | null | null | 2 | 3 | 75 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
165 | GREEN-077 | Structure of Sets with Bounded Representation | Suppose $A \subset [N]$ has size $\geq c\sqrt{N}$ and representation function $r_A(n) \leq r$ for all $n$. What can be said about the structure of $A$? | Sets with bounded representation function (few ways to write sums) have special structure. Understanding this structure connects Sidon set theory with additive bases.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Curre... | 1 | partially_solved | null | null | 2 | 3 | 70 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
167 | GREEN-079 | Disjoint Sumsets Construction | For arbitrarily large $n$, does there exist an abelian group $H$ with $|H| = n^{2+o(1)}$ and subsets $A_1, \dots, A_n, B_1, \dots, B_n$ satisfying $|A_i||B_i| \geq n^{2-o(1)}$, $|A_i + B_i| = |A_i||B_i|$, such that $A_i + B_i$ are pairwise disjoint from $A_j + B_k$ ($j \neq k$)? | This asks if one can partition a group into many disjoint sumsets with no doubling. It connects to the structure of Sidon sets and extremal problems in additive combinatorics.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current liter... | 2 | open | null | null | 2 | 3 | 69 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
168 | GREEN-080 | Cap Sets in F_7^n | What is the largest subset $A \subset \mathbb{F}_7^n$ for which $A - A$ intersects $\{-1, 0, 1\}^n$ only at 0? | This is a cap set problem in $\mathbb{F}_7^n$ with restricted difference set. Recent polynomial method breakthroughs dramatically improved bounds for $\mathbb{F}_3^n$, but $\mathbb{F}_7$ remains open.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classific... | 2 | partially_solved | null | null | 2 | 3 | 73 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
169 | GREEN-081 | Covering by Random Translates | If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random with $|A| = \sqrt{p}$, can we almost surely cover $\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$ translates of $A$? | This asks about the covering properties of random sets. Understanding when random sets form good coverings connects probability, additive combinatorics, and coding theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literatur... | 1 | open | null | null | 2 | 3 | 68 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
170 | GREEN-082 | Hamming Ball Covering Growth | Let $r$ be fixed and let $H(r)$ be the Hamming ball of radius $r$ in $\mathbb{F}_2^n$. Let $f(r)$ be the smallest constant such that there exist infinitely many $n$ with subspaces $V_n \leq \mathbb{F}_2^n$ satisfying $V_n + H(r) = \mathbb{F}_2^n$ and $|V_n| = (f(r) + o(1)) \frac{2^n}{|H(r)|}$. Does $f(r) \to \infty$? | This asks if covering $\mathbb{F}_2^n$ by Hamming ball translates requires increasingly inefficient packings as $r$ grows. It connects coding theory with the geometry of finite vector spaces.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** ... | 2 | partially_solved | null | null | 2 | 3 | 66 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
171 | GREEN-083 | Pyjama Set Covering | How many rotated (about the origin) copies of the "pyjama set" $\{(x, y) \in \mathbb{R}^2 : \operatorname{dist}(x, \mathbb{Z}) \leq \varepsilon\}$ are needed to cover $\mathbb{R}^2$? | The pyjama set is a union of vertical strips. This beautiful geometric problem, solved by Manners (2015), asks how many rotations are needed to cover the plane. It connects geometry, combinatorics, and Fourier analysis.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_s... | 1 | partially_solved | null | null | 6 | 3 | 74 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
172 | GREEN-084 | Cohn-Elkies Scheme for Circle Packings | Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings? | Cohn-Elkies developed a linear programming approach that proved optimal sphere packing in dimensions 8 and 24. Whether their method extends to circles in the plane remains a major open question in discrete geometry.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classi... | 2 | open | null | null | 6 | 3 | 71 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
173 | GREEN-085 | Covering by Residue Classes | Let $N$ be large. For each prime $p$ with $N^{0.51} \leq p < 2N^{0.51}$, pick a residue $a(p) \in \mathbb{Z}/p\mathbb{Z}$. Is $\#\{n \in [N] : n \equiv a(p) \pmod p \text{ for some } p\} \gg N^{1-o(1)}$? | This asks if residue classes from medium-sized primes nearly cover $[N]$. It connects sieve theory with covering problems and the distribution of primes.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No... | 1 | open | null | null | 1 | 3 | 69 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | [Discussion #4 literature review, 2026-10-08] `10.1017/S0308210517000144` → `10.1017/S0308210517000191`. | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
174 | GREEN-086 | Sieving by Many Small Primes | Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leq p_1 < p_2 < \dots < p_{1000} < N^{9/10}$. Does the remaining set have size at most $\frac{1}{10}N$? | This asks whether aggressive sieving by many small primes can remove most of $[N]$. Understanding sieve limits is fundamental in analytic number theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature ... | 1 | partially_solved | null | null | 1 | 3 | 67 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
175 | GREEN-087 | Residue Class Multiple Coverage | Can we pick residue classes $a_p \pmod p$, one for each prime $p \leq N$, such that every integer $\leq N$ lies in at least 10 of them? | This asks if we can achieve high-multiplicity covering using one residue class per prime. It's dual to sieving problems and connects to the large sieve.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The... | 1 | open | null | null | 1 | 3 | 68 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
176 | GREEN-088 | Maximal Covering Interval | What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \pmod p$, one for each prime $p \leq x$? | This is the classical covering problem in sieve theory. Determining the optimal relationship between $x$ and $y$ would have significant implications for understanding the distribution of primes.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:... | 2 | partially_solved | null | null | 1 | 3 | 70 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
177 | GREEN-089 | Random Walk Mixing on Alternating Groups | Pick $x_1, \dots, x_k \in A_n$ at random. Is it true that, almost surely as $n \to \infty$, the random walk on this set of generators and their inverses equidistributes in time $O(n \log n)$? | This asks about mixing time for random walks on the alternating group with random generators. Determining optimal mixing times connects probability, group theory, and spectral graph theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PA... | 2 | partially_solved | null | null | 4 | 3 | 69 | 3 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
178 | GREEN-090 | Bounds for Approximate Group Classification | Find bounds in the classification theorem for approximate groups. | The Breuillard-Green-Tao classification shows approximate groups resemble actual groups. However, the bounds in this theorem are extremely poor. Improving them would have significant applications in additive combinatorics and geometric group theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-0... | 2 | partially_solved | null | null | 4 | 3 | 72 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,097 | GREEN-097 | N-Queens Problem Asymptotics | In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \times n$ chessboard? | The n-queens problem asks for asymptotic formulas for the number of ways to place n non-attacking queens on an n×n board. Recent work has made progress on both upper and lower bounds, but the precise asymptotic behavior remains elusive.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Stat... | 1 | partially_solved | null | null | 2 | 3 | 145 | 9 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | false | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | Exact duplicate of GREEN-061 in Ben Green's Open Problems | [
"GREEN-061"
] | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,098 | GREEN-098 | Bounds for Homogeneous Polynomial Zeros | Let $d \geq 3$ be an odd integer. Give bounds on $\nu(d)$ such that if $n > \nu(d)$ the following is true: given any homogeneous polynomial $F(\mathbf{x}) \in \mathbb{Z}[x_1, \dots, x_n]$ of degree $d$, there is some $\mathbf{x} \in \mathbb{Z}^n \setminus \{\mathbf{0}\}$ such that $F(\mathbf{x}) = 0$. | This asks for explicit bounds on how many variables are needed to guarantee integer zeros of homogeneous polynomials. Classical results give existence but quantitative bounds remain challenging, especially for higher degrees.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** parti... | 2 | partially_solved | null | null | 1 | 3 | 78 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | false | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | Exact duplicate of GREEN-062 in Ben Green's Open Problems | [
"GREEN-062"
] | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,099 | GREEN-099 | Polynomial Solutions in Dense Sets | Finding a single solution to a polynomial equation $F(x_1, \dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of such solutions in $A$ is roughly $\alpha^n$ times the number of solutions in $[X]$? | This problem asks when dense sets contain the "expected" number of polynomial solutions. Understanding density conditions that guarantee proportional solution counts connects number theory with additive combinatorics.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_sol... | 2 | partially_solved | null | null | 1 | 3 | 71 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | false | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | Exact duplicate of GREEN-063 in Ben Green's Open Problems | [
"GREEN-063"
] | null | null | null | null | null | null | null | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,100 | GREEN-100 | Sofic Groups | Is every group well-approximated by finite groups? | A group is sofic if it can be approximated by finite symmetric groups in a precise sense. Whether all groups are sofic is a major open question in group theory with connections to dynamics, graph theory, and combinatorics.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-09-27)
**Status:** solved
*... | 2 | partially_solved | null | null | 4 | 3 | 92 | 6 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | false | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "partial",
"evidence": "unrefereed_claim"
} | Exact duplicate of GREEN-064 in Ben Green's Open Problems | [
"GREEN-064"
] | null | null | null | Reconciled with the existing literature review of 2026-08-17: The universal soficity assertion in the database background was refuted in August 2026 by an explicit nonsofic group, with an independent expert preprint already deriving a torsion-free example from the same key criterion. This is a consistency correction ba... | [
{
"url": "https://cdn.openai.com/pdf/ten-proofs-oai.pdf",
"label": "OpenAI, Nonsofic groups exist, Chapter 3 of Ten Advances in Mathematics and Theoretical Computer Science, updated 2026-08-06."
},
{
"url": "https://arxiv.org/abs/2608.02025",
"label": "Francesco Fournier-Facio, A torsion-free no... | 2026-09-27T00:00:00 | OpenAI, Nonsofic groups exist, Chapter 3 of Ten Advances in Mathematics and Theoretical Computer Science, updated 2026-08-06.: https://cdn.openai.com/pdf/ten-proofs-oai.pdf
Francesco Fournier-Facio, A torsion-free non-sofic group, arXiv:2608.02025 (2026-08-03).: https://arxiv.org/abs/2608.02025
Ben Green, 100 Open Prob... | https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,102 | ALG-004 | Hadamard Conjecture | For every positive integer $k$, does there exist a Hadamard matrix of order $4k$? | A Hadamard matrix is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. The Hadamard conjecture, dating back to 1893, states that such matrices exist for all orders that are multiples of 4. These matrices have important applications in coding theory, signal processing, and quantum... | 4 | open | null | null | 4 | null | 387 | 31 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | ALG-002 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,103 | ALG-050 | Köthe Conjecture | If a ring has no nil ideal other than $\{0\}$, does it follow that it has no nil one-sided ideal other than $\{0\}$? | The Köthe conjecture, proposed by Gottfried Köthe in 1930, is a fundamental problem in ring theory concerning the structure of nil ideals. A nil ideal is one in which every element is nilpotent. The conjecture asks whether the absence of two-sided nil ideals implies the absence of one-sided nil ideals. Despite being st... | 4 | partially_solved | null | null | 4 | null | 245 | 18 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | false | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "partial",
"evidence": "unrefereed_claim"
} | Internal Seed duplicate of ALG-003 (id=38, 'The Köthe Conjecture') | [
"ALG-003"
] | ALG-003 | null | null | Adamczewski, Böhmler and Marczinzik announce a counterexample to Köthe’s conjecture (September 2026). Recorded as a claimed negative resolution in an unrefereed preprint.
**What remains.** Independent expert verification and peer review of the recent counterexample remain. | [
{
"url": "https://arxiv.org/abs/2609.07996",
"label": "Tom Adamczewski, Bernhard Böhmler and Rene Marczinzik, A counterexample to Köthe’s conjecture and a question of Rowen"
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Al... | 2026-09-27T00:00:00 | Tom Adamczewski, Bernhard Böhmler and Rene Marczinzik, A counterexample to Köthe’s conjecture and a question of Rowen: https://arxiv.org/abs/2609.07996 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,104 | ALG-005 | Connes Embedding Problem | Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor? | The Connes embedding problem, formulated by Alain Connes in 1976, is a central question in the theory of von Neumann algebras. In 2020, Ji, Natarajan, Vidick, Wright, and Yuen published a paper claiming to have shown the problem has a negative answer, based on connections to quantum complexity theory and the equivalenc... | 5 | solved | null | null | 4 | null | 312 | 28 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | [
"ALG-057"
] | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "resolution",
"evidence": "published"
} | null | null | ALG-004 | null | null | Ji, Natarajan, Vidick, Wright and Yuen refute Connes embedding through MIP*=RE (2020).
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://arxiv.org/abs/2001.04383",
"label": "Ji et al., MIP*=RE"
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates and follow-ups"
}
] | 2026-09-27T00:00:00 | Ji et al., MIP*=RE: https://arxiv.org/abs/2001.04383 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,105 | ALG-006 | Jacobson's Conjecture | For a left-and-right Noetherian ring $R$, is the intersection of all powers of the Jacobson radical $J(R)$ equal to zero? | Jacobson's conjecture addresses a fundamental question about the structure of Noetherian rings. The Jacobson radical of a ring consists of elements that annihilate all simple modules, and understanding its intersection over all powers relates to the ring's nilpotent elements and its representation theory. While the con... | 4 | partially_solved | null | null | 4 | null | 198 | 14 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | ALG-005 | [Discussion #4 literature review, 2026-10-08] . S. Joka's [arXiv:2601.13475](https://arxiv.org/abs/2601.13475) is listed as a claimed unconditional proof; the author's comment on v5 (31 May 2026) says that the proof is not correct. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,106 | ALG-007 | Zauner's Conjecture | Do SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite dimensions? | Zauner's conjecture, proposed in 1999, concerns the existence of a special type of quantum measurement in Hilbert spaces of all finite dimensions. A SIC-POVM consists of d² unit vectors in a d-dimensional complex Hilbert space that are equiangular - the absolute inner product of any two distinct vectors is constant. Th... | 4 | partially_solved | null | null | 4 | null | 176 | 16 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | [
"ALG-052"
] | null | null | null | ALG-006 | [Discussion #4 literature review, 2026-10-08] . S. Joka's [arXiv:2601.13475](https://arxiv.org/abs/2601.13475) is listed as a claimed unconditional proof; the author's comment on v5 (31 May 2026) says that the proof is not correct. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,107 | ALG-008 | Casas-Alvero Conjecture | If a univariate polynomial $f$ of degree $d$ over a field of characteristic 0 shares a common factor with each of its first $d-1$ derivatives, must $f$ be a power of a linear polynomial? | The Casas-Alvero conjecture, proposed in 2001, connects the factorization of a polynomial with the factorization of its derivatives. If $f(x)$ has degree $d$ and for each $k = 1, 2, \ldots, d-1$, the polynomial $f(x)$ shares a root with its $k$-th derivative $f^{(k)}(x)$, the conjecture states that $f$ must be of the f... | 3 | partially_solved | null | null | 4 | null | 154 | 11 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | [
"ALG-056"
] | null | null | null | ALG-007 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,108 | ALG-009 | Andrews-Curtis Conjecture | Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugations of relators? | The Andrews-Curtis conjecture, proposed in 1965, is a central problem in combinatorial group theory. A balanced presentation has an equal number of generators and relators. The conjecture asks whether such presentations of the trivial group can always be simplified to the form $\langle x_1, \ldots, x_n \mid x_1, \ldots... | 4 | partially_solved | null | null | 4 | null | 212 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | [
"ALG-023",
"ALG-053"
] | null | null | null | ALG-008 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,109 | ALG-010 | Bounded Burnside Problem | For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2,5)$ finite? | The Bounded Burnside problem asks which free Burnside groups are finite. A free Burnside group $B(m,n)$ is the largest group with $m$ generators in which every element has order dividing $n$. It is known that $B(m,n)$ is finite for $n \in \{2, 3, 4, 6\}$ and for certain other special values, and infinite for most large... | 4 | partially_solved | null | null | 4 | null | 189 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | [
"ALG-024"
] | null | null | null | ALG-009 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,110 | ALG-011 | Herzog-Schönheim Conjecture | If a finite system of left cosets of subgroups of a group $G$ partitions $G$, then must at least two of the subgroups have the same index in $G$? | The Herzog-Schönheim conjecture, proposed in 1974, concerns coset decompositions of groups. If $G$ is partitioned by cosets $g_1H_1, g_2H_2, \ldots, g_kH_k$ where each $H_i$ is a subgroup of $G$ and the cosets are pairwise disjoint, the conjecture states that at least two of the indices $[G:H_i]$ must be equal. While p... | 3 | partially_solved | null | null | 4 | null | 142 | 12 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | [
"ALG-026",
"ALG-054"
] | null | null | null | ALG-010 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,112 | ALG-012 | Existence of Perfect Cuboids | Does there exist a rectangular cuboid where all edges, face diagonals, and space diagonals have integer lengths? | A perfect cuboid (also called a perfect box or Euler brick with space diagonal) would be a rectangular parallelepiped with integer edge lengths $a$, $b$, $c$ such that the face diagonals $\sqrt{a^2+b^2}$, $\sqrt{b^2+c^2}$, $\sqrt{a^2+c^2}$ and the space diagonal $\sqrt{a^2+b^2+c^2}$ are all integers. Despite extensive ... | 3 | open | null | null | 4 | null | 234 | 21 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | [Discussion #4 literature review, 2026-10-08] [arXiv:2008.06045](https://arxiv.org/abs/2008.06045) is by A. Pokrovskiy (cited: Pokrovskiy, Sudakov and Yepremyan). | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,114 | ALG-014 | McKay Conjecture | For a finite group $G$ and prime $p$, is the number of irreducible complex characters of $G$ whose degree is not divisible by $p$ equal to the corresponding number for the normalizer of a Sylow $p$-subgroup? | The McKay conjecture, proposed in the 1970s, is a central problem in the representation theory of finite groups. It predicts a surprising relationship between the character degrees of a group and those of a much smaller subgroup (the normalizer of a Sylow $p$-subgroup). The conjecture has been verified for many importa... | 4 | solved | null | null | 4 | null | 156 | 13 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "resolution",
"evidence": "published"
} | null | null | null | null | null | Cabanes and Späth prove the McKay character-degree equality for every finite group and prime. The arXiv record lists the paper as to appear in Annals of Mathematics.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://arxiv.org/abs/2410.20392",
"label": "Marc Cabanes and Britta Späth, The McKay Conjecture on character degrees (2024; revised 2025)"
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status upda... | 2026-09-27T00:00:00 | Marc Cabanes and Britta Späth, The McKay Conjecture on character degrees (2024; revised 2025): https://arxiv.org/abs/2410.20392 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,115 | ALG-015 | Are All Groups Surjunctive? | Is every group surjunctive? That is, for any group $G$, if $\phi: A^G \to A^G$ is a cellular automaton that is injective, must it also be surjective? | A group $G$ is called surjunctive if every injective cellular automaton on $G$ is automatically surjective. Equivalently, this asks whether the dynamical system defined by a cellular automaton on the group can be injective without being bijective. Gromov and Weiss proved that all sofic groups are surjunctive, and all k... | 4 | open | null | null | 4 | null | 143 | 11 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,119 | NT-016 | Catalan-Mersenne Conjecture | Are all Catalan-Mersenne numbers $C_n$ composite for $n > 4$? Here $C_0 = 2$ and $C_{n+1} = 2^{C_n} - 1$. | The Catalan-Mersenne conjecture concerns a doubly exponential sequence where each term is a Mersenne number with exponent equal to the previous term. The sequence grows extraordinarily rapidly: $C_0 = 2$, $C_1 = 3$, $C_2 = 7$, $C_3 = 127$, $C_4 = 170141183460469231731687303715884105727$ (a 39-digit number). The first f... | 4 | open | null | null | 1 | null | 287 | 24 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,120 | NT-017 | Are There Infinitely Many Mersenne Primes? | Are there infinitely many prime numbers of the form $2^p - 1$ where $p$ is prime? | Mersenne primes are primes of the form $M_p = 2^p - 1$ where $p$ is itself prime. As of 2024, only 51 Mersenne primes are known, the largest being $2^{82589933} - 1$ discovered in 2018. Despite their rarity, it is conjectured that infinitely many exist. Mersenne primes are intimately connected to perfect numbers throug... | 5 | open | null | null | 1 | null | 567 | 49 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | false | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | Internal Seed duplicate of NT-007 (id=26, 'Are there infinitely many Mersenne primes?') | [
"NT-007"
] | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,121 | GEO-001 | Sphere Packing Problem in Higher Dimensions | What is the densest packing of spheres in dimensions 4 through 23? More generally, what is the optimal sphere packing density in dimension $n$? | The sphere packing problem asks for the densest arrangement of non-overlapping spheres in $n$-dimensional space. In dimension 3, Kepler's conjecture (proved by Hales in 1998) shows the densest packing has density $\pi/\sqrt{18} \approx 0.7405$. In 2016, Maryna Viazovska proved that the E₈ lattice gives the densest pack... | 5 | partially_solved | null | null | 6 | null | 398 | 34 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | false | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | Internal Seed duplicate of GEO-002 (id=19, 'Sphere Packing in Higher Dimensions') | [
"GEO-002"
] | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,122 | GEO-006 | Mahler's Conjecture | Among all centrally symmetric convex bodies in $\mathbb{R}^n$, does the cube (or cross-polytope) minimize the product of the body's volume and the volume of its polar dual? | Mahler's conjecture, proposed in 1939, concerns a fundamental geometric quantity called the Mahler volume, defined as the product of a convex body's volume with the volume of its polar dual. Kurt Mahler conjectured that among all centrally symmetric convex bodies in $\mathbb{R}^n$, this product is minimized by the cube... | 4 | partially_solved | null | null | 6 | null | 245 | 21 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | GEO-002 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,123 | GEO-007 | The Illumination Conjecture | Can every convex body in $n$-dimensional space be illuminated by at most $2^n$ point light sources? | The illumination conjecture, also known as Hadwiger's covering conjecture in one of its forms, asks whether every convex body in $\mathbb{R}^n$ can be illuminated by at most $2^n$ point light sources placed outside the body. A point on the surface is considered illuminated if the ray from the light source to that point... | 4 | open | null | null | 6 | null | 187 | 16 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | [
"GEOM-008"
] | null | null | null | GEO-003 | [Discussion #4 literature review, 2026-10-08] arXiv:2502.17651 is "METAL: A Multi-Agent Framework for Chart Generation with Test-Time Scaling" by B. Li et al.; cited paper (Wang–Zahl, Kakeya in three dimensions): arXiv:2502.17655. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,124 | GEO-008 | Kakeya Needle Problem | What is the minimum area of a region in the plane in which a unit line segment can be continuously rotated through 360 degrees? | The Kakeya needle problem asks for the smallest area set in the plane within which a unit line segment can be rotated continuously through 360 degrees, returning to its initial position. While Besicovitch showed in 1928 that there exist Kakeya sets of arbitrarily small positive measure, the question of what happens whe... | 4 | partially_solved | null | null | 6 | null | 312 | 27 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | GEO-004 | [Discussion #4 literature review, 2026-10-08] [arXiv:2407.02587](https://arxiv.org/abs/2407.02587) is by Z. Deng (cited: J. Baek; Baek's paper on the optimality of Gerver's sofa is arXiv:2411.19826). | null | Reconciled with the existing literature review of 2026-08-17: The infimum area is zero: Besicovitch constructed planar Kakeya sets of measure zero. This is a consistency correction based on that cited review; the discussion does not independently re-verify its proof claims.
**What remains.** No work remains for the li... | [
{
"url": "https://en.wikipedia.org/wiki/Kakeya_set",
"label": "Kakeya set overview (accessed 2026-08-17)."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates and follow-ups"
}
] | 2026-09-27T00:00:00 | Kakeya set overview (accessed 2026-08-17).: https://en.wikipedia.org/wiki/Kakeya_set | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,125 | GEO-009 | Bellman's Lost in a Forest Problem | What is the shortest path that guarantees escape from a forest of known shape and size, starting from an unknown location? | Bellman's lost in a forest problem asks for the shortest universal path that guarantees reaching the boundary of a region, regardless of starting position and orientation. For a circular forest of radius 1, the problem was solved by various authors with a path of length approximately 7.2898. However, for other shapes l... | 3 | partially_solved | null | null | 6 | null | 198 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | [
"GEOM-025"
] | null | null | null | GEO-005 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,131 | COMB-013 | The Union-Closed Sets Conjecture | For any finite family of finite sets that is closed under taking unions, must there exist an element that belongs to at least half of the sets? | The union-closed sets conjecture, also known as Frankl's conjecture after Peter Frankl who popularized it in 1979, is a simple-to-state problem in extremal combinatorics. A family of sets is union-closed if the union of any two sets in the family is also in the family. The conjecture asserts that in any non-trivial uni... | 4 | partially_solved | null | null | 2 | null | 334 | 28 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | false | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | Internal Seed duplicate of COMB-005 (id=55, 'Frankl's Union-Closed Sets Conjecture') | [
"COMB-005"
] | COMB-003 | [Discussion #4 literature review, 2026-10-08] , `COMB-003` (1321): arXiv:1704.07022 is "Note on the union-closed sets conjecture" by A. Raz; cited paper (Reimer, "An average set size theorem", Combin. Probab. Comput. 12 (2003)): `10.1017/S0963548302005230`. | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,132 | COMB-014 | Singmaster's Conjecture | Does there exist a finite upper bound on how many times a number (other than 1) can appear in Pascal's triangle? | Singmaster's conjecture, proposed by David Singmaster in 1971, concerns the frequency of entries in Pascal's triangle. While 1 appears infinitely often (along the edges), and 2 appears exactly three times, larger numbers can appear multiple times in different positions. For example, 120 appears six times. The conjectur... | 3 | partially_solved | null | null | 2 | null | 298 | 26 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | false | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | Duplicate of canonical problem EP-849 | [
"EP-849"
] | COMB-004 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,135 | SET-001 | The Continuum Hypothesis | Is there a set whose cardinality is strictly between that of the integers and the real numbers? | The continuum hypothesis (CH), proposed by Georg Cantor in 1878, states that there is no set with cardinality strictly between that of the integers and the real numbers. Equivalently, it asserts that the cardinality of the continuum (the real numbers) is $\aleph_1$, the second smallest infinite cardinal. Gödel proved i... | 5 | solved | null | null | 10 | null | 623 | 54 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | false | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "resolution",
"evidence": "published"
} | Duplicate of canonical problem HIL-001 | [
"HIL-001"
] | null | null | null | Independent of ZFC, assuming ZFC is consistent, by Gödel and Cohen. “Solved” records independence; it does not assert the continuum hypothesis or its negation.
**What remains.** Neither direction is provable in ZFC if ZFC is consistent. | [
{
"url": "https://doi.org/10.1073/pnas.24.12.556",
"label": "Kurt Godel, The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis, Proceedings of the National Academy of Sciences 24 (1938), 556-557."
},
{
"url": "https://doi.org/10.1073/pnas.50.6.1143",
"label": "Paul J... | 2026-09-27T00:00:00 | Kurt Godel, The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis, Proceedings of the National Academy of Sciences 24 (1938), 556-557.: https://doi.org/10.1073/pnas.24.12.556
Paul J. Cohen, The Independence of the Continuum Hypothesis, Proceedings of the National Academy of Sciences 50 (196... | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,137 | NT-019 | Are There Infinitely Many Sophie Germain Primes? | Are there infinitely many primes $p$ such that $2p + 1$ is also prime? | A Sophie Germain prime is a prime $p$ where $2p+1$ is also prime. These primes are named after French mathematician Sophie Germain, who used them in her work on Fermat's Last Theorem. Examples include 2, 3, 5, 11, 23, and 29. The conjecture that infinitely many exist is closely related to the twin prime conjecture and ... | 5 | open | null | null | 1 | null | 389 | 33 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,138 | AG-005 | The Hodge Conjecture | On a projective algebraic variety, is every Hodge class a rational linear combination of classes of algebraic cycles? | The Hodge conjecture is one of the seven Millennium Prize Problems, with a $1 million prize for its solution. Proposed by William Hodge in 1950, it concerns the deep relationship between the topology and algebraic geometry of complex projective varieties. In simple terms, it asks whether certain topological cycles (Hod... | 5 | partially_solved | null | null | 5 | null | 534 | 46 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | false | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | Duplicate of canonical problem MPP-006 | [
"MPP-006"
] | AG-001 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,140 | AG-003 | The Birch and Swinnerton-Dyer Conjecture | For an elliptic curve $E$ over the rationals, does the rank of its group of rational points equal the order of vanishing of its $L$-function at $s=1$? | The Birch and Swinnerton-Dyer (BSD) conjecture is one of the seven Millennium Prize Problems. It connects the arithmetic properties of elliptic curves (specifically, the group of rational points) with analytic properties (the behavior of the associated $L$-function). The conjecture predicts a precise relationship betwe... | 5 | partially_solved | null | null | 5 | null | 687 | 59 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | false | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | Duplicate of canonical problem MPP-005 | [
"MPP-005"
] | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,141 | DYN-001 | The Weinstein Conjecture | Does every Reeb vector field on a closed contact manifold have at least one periodic orbit? | The Weinstein conjecture, proposed by Alan Weinstein in 1978, is a fundamental problem in symplectic geometry and dynamical systems. A Reeb vector field is a special type of vector field on a contact manifold, and the conjecture predicts the existence of closed orbits under very general conditions. The conjecture was p... | 4 | partially_solved | null | null | 11 | null | 276 | 23 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | [
"DYN-005"
] | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,142 | DYN-002 | The Painlevé Conjecture | In the $n$-body problem with $n \geq 4$, can non-collision singularities occur in finite time? | The Painlevé conjecture concerns the $n$-body problem in celestial mechanics, asking whether the motion of $n$ point masses under gravitational attraction can develop a singularity (infinite velocities or unbounded positions) in finite time without any collisions occurring. For $n=3$, Sundman proved in 1912 that non-co... | 5 | solved | null | null | 11 | null | 298 | 25 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 11,
"name": "dynamical_systems",
"display_name": "Dynamical Systems",
"description": "Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.",
"slug": "dynamical-systems",
"order_index": 11,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "resolution",
"evidence": "published"
} | null | null | null | null | null | Noncollision singularities exist: Xia constructed five-body examples (1992), and Xue proved the planar four-body result (Acta Mathematica 224, 2020).
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220228202444113958904",
"label": "J. Xue, Noncollision Singularities in a Planar Four-body Problem, Acta Math. 224 (2020), 253--388."
},
{
"url": "https://www.math.harvard.edu/media/CDM_Xue.pdf",
"label": "On Painleve Conjecture, sur... | 2026-09-27T00:00:00 | J. Xue, Noncollision Singularities in a Planar Four-body Problem, Acta Math. 224 (2020), 253--388.: https://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220228202444113958904
On Painleve Conjecture, survey/preprint by J. Xue (accessed 2026-08-17).: https://www.math.harvard.edu/media/CDM_Xue.pdf | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,144 | GT-008 | Cereceda's Conjecture | For any $k$-chromatic graph, can its $k$-colorings be transformed into each other by recoloring one vertex at a time, staying within $k$ colors, in polynomial time in the number of vertices? | Cereceda's conjecture concerns the diameter of the reconfiguration graph of $k$-colorings. Given a graph $G$ with chromatic number $k$, consider the graph whose vertices are all proper $k$-colorings of $G$, with two colorings adjacent if they differ on exactly one vertex. The conjecture, proposed in 2007, states that t... | 3 | partially_solved | null | null | 3 | null | 198 | 17 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | [
"GRAPH-061"
] | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "scope_correction",
"evidence": "scope_review"
} | null | null | null | null | null | The stored k-coloring claim fails even for K2. The standard Cereceda conjecture concerns (d+2)-colorings of d-degenerate graphs and a quadratic bound; a counterexample to the imported wording does not resolve it.
**What remains.** The earlier solved label is not accepted for the intended or insufficiently specified pr... | [
{
"url": "https://doi.org/10.1016/j.disc.2007.07.028",
"label": "L. Cereceda, J. van den Heuvel and M. Johnson, Connectedness of the graph of vertex-colourings, Discrete Mathematics 308 (2008), 913--919, doi:10.1016/j.disc.2007.07.028."
},
{
"url": "https://arxiv.org/abs/1903.05619",
"label": "N... | 2026-09-27T00:00:00 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,146 | TOP-003 | The Whitehead Conjecture | Is every aspherical closed manifold whose fundamental group has no non-trivial perfect normal subgroups a $K(\pi, 1)$ space? | The Whitehead conjecture, posed by J.H.C. Whitehead, concerns a special class of topological spaces. A space is aspherical if all its homotopy groups above dimension 1 vanish, and it is a $K(\pi, 1)$ if it is aspherical and connected. The conjecture asks whether certain algebraic conditions on the fundamental group gua... | 4 | partially_solved | null | null | 7 | null | 234 | 20 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | [
"TOP-016"
] | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "scope_correction",
"evidence": "scope_review"
} | null | null | null | [Discussion #4 literature review, 2026-10-08] , `TOP-003` (1333): `10.4171/QT/82` is a paper by Sabloff–Sullivan; cited paper: `10.4171/QT/83`. | null | The stored question is tautological: an aspherical connected manifold is a K(pi,1). This does not settle the classical Whitehead conjecture about subcomplexes of aspherical two-dimensional complexes.
**What remains.** The earlier solved label is not accepted for the intended or insufficiently specified problem. The ci... | [
{
"url": "https://pi.math.cornell.edu/~hatcher/AT/AT.pdf",
"label": "Allen Hatcher, Algebraic Topology, Section 1.B, online textbook."
},
{
"url": "https://doi.org/10.4213/im9597",
"label": "A. M. Mikhovich, Rational and p-adic analogues of J. H. C. Whitehead's conjecture, Izvestiya: Mathematics... | 2026-09-27T00:00:00 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,149 | GEO-010 | The Knaster Problem | Can a solid cube be completely covered by finitely many smaller homothetic cubes with ratio less than 1, such that the interiors are disjoint? | The Knaster problem, also known as the cube packing problem, asks whether a unit cube can be covered by finitely many non-overlapping smaller cubes, each similar to the original with ratio $< 1$. In 1979, Mycielski proved this is impossible in dimension 2 (for squares), but the 3-dimensional case remains open. The prob... | 4 | partially_solved | null | null | 6 | null | 189 | 16 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "scope_correction",
"evidence": "scope_review"
} | null | null | GEO-006 | null | null | Subdividing a cube answers the literal tiling question, but this is not a resolution of the intended Knaster problem; its source statement needs recovery.
**What remains.** The earlier solved label is not accepted for the intended or insufficiently specified problem. The cited partial result or literal observation is ... | [
{
"url": "https://en.wikipedia.org/wiki/Cube",
"label": "Elementary cube subdivision argument (checked 2026-08-17)."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates and follow-ups"
}
] | 2026-09-27T00:00:00 | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,153 | NT-022 | Polignac's Conjecture | For every even number $n$, are there infinitely many pairs of consecutive primes differing by $n$? | Polignac's conjecture, proposed by Alphonse de Polignac in 1849, is a vast generalization of the twin prime conjecture. It asserts that for every even integer $n$, there exist infinitely many prime gaps of exactly size $n$. The twin prime conjecture is the special case $n = 2$. While Zhang's 2013 breakthrough showed in... | 5 | partially_solved | null | null | 1 | null | 389 | 33 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | [
"OPG-37289"
] | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,156 | ALG-016 | The Babai Conjecture on Graph Isomorphism | Can graph isomorphism be decided in quasi-polynomial time for all graphs? | The Babai conjecture concerns the computational complexity of determining whether two graphs are isomorphic. In 2015, László Babai announced a quasi-polynomial time algorithm for graph isomorphism (running in time $2^{O(\log^c n)}$ for some constant $c$), improving on the previous best bound. While a flaw was found in ... | 4 | solved | null | null | 4 | null | 445 | 38 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "resolution",
"evidence": "published"
} | null | null | null | null | null | Babai’s quasipolynomial-time graph-isomorphism algorithm answers the stated complexity question affirmatively.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://arxiv.org/abs/1512.03547",
"label": "László Babai, Graph Isomorphism in Quasipolynomial Time (STOC 2016)"
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun, status updates and follow-ups"
}
] | 2026-09-27T00:00:00 | László Babai, Graph Isomorphism in Quasipolynomial Time (STOC 2016): https://arxiv.org/abs/1512.03547 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,157 | NT-023 | Pillai's Conjecture | For each positive integer $k$, does the equation $|2^m - 3^n| = k$ have only finitely many solutions in positive integers $m$ and $n$? | Pillai's conjecture, proposed by Subbayya Sivasankaranarayana Pillai, concerns the gaps between powers of 2 and powers of 3. The conjecture generalizes to any two multiplicatively independent integers $a$ and $b$: the equation $|a^m - b^n| = k$ should have only finitely many solutions for each fixed $k$. This is relate... | 4 | partially_solved | null | null | 1 | null | 198 | 17 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | [
"AMR-093-0110"
] | {
"source_url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"checked_at": "2026-09-27",
"kind": "partial",
"evidence": "unrefereed_claim"
} | null | null | null | [Discussion #4 literature review, 2026-10-08] `10.1007/BF01388636` is a paper by Rohrlich; cited paper: `10.1007/BF01388644`. | null | Reconciled with the existing literature review of 2026-08-17: For fixed k, the equation 2^m-3^n=plus/minus k has only finitely many solutions by the finiteness theorem for S-unit equations. This is a consistency correction based on that cited review; the discussion does not independently re-verify its proof claims.
**... | [
{
"url": "https://doi.org/10.1007/BF01388636",
"label": "J.-H. Evertse, On equations in S-units and the Thue-Mahler equation, Invent. Math. 75 (1984), 561--584."
},
{
"url": "https://huggingface.co/datasets/ulamai/UnsolvedMath/discussions/4",
"label": "UnsolvedMath discussion #4 — Alper Ferudun,... | 2026-09-27T00:00:00 | J.-H. Evertse, On equations in S-units and the Thue-Mahler equation, Invent. Math. 75 (1984), 561--584.: https://doi.org/10.1007/BF01388636 | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,158 | NT-024 | Erdős-Straus Conjecture | For every integer $n \geq 2$, can $\frac{4}{n}$ be expressed as the sum of three unit fractions $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$? | The Erdős-Straus conjecture asks whether every fraction $4/n$ (for $n \geq 2$) can be written as a sum of three unit fractions (fractions with numerator 1). The conjecture has been verified computationally for all $n$ up to $10^{17}$ and proven for several infinite families of values, but the general case remains open.... | 3 | open | null | null | 1 | null | 289 | 24 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | false | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | Duplicate of canonical problem EP-242 | [
"EP-242"
] | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,159 | NT-025 | The Gauss Circle Problem | What is the optimal error term in the formula for the number of lattice points inside a circle of radius $r$? | The Gauss circle problem asks for the number of integer lattice points $(x,y)$ satisfying $x^2 + y^2 \leq r^2$. The main term is $\pi r^2$ (the area of the circle), but determining the optimal error term has been a central problem in analytic number theory for over 150 years. It is known that the error is $O(r^{2/3})$ ... | 5 | partially_solved | null | null | 1 | null | 367 | 31 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | [
"NT-032"
] | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
1,160 | ALG-017 | Birch-Tate Conjecture | Does the order of the center of the Steinberg group of the ring of integers of a number field relate to the value of the Dedekind zeta function at $s=-1$? | The Birch-Tate conjecture concerns the relationship between algebraic K-theory and special values of zeta functions. Specifically, it relates the order of $K_2$ of the ring of integers of a number field to the value of the Dedekind zeta function at $s = -1$. This conjecture is part of a broader program connecting algeb... | 5 | partially_solved | null | null | 4 | null | 178 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | [
"ALG-051",
"ALG-055"
] | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.