Consider a set $X$ with $|X| = n\geq 1$ elements. A family $\mathcal{F}$ of distinct subsets of $X$ is said to have property $\mathcal{P}$ if there exist $A,B \in \mathcal{F}$ so that $A\subset B$ and $|B\setminus A| = 1$.

i) Determine the least value $m$, so that any family $\mathcal{F}$ with $|\mathcal{F}| > m$ has property $\mathcal{P}$.
ii) Describe all families $\mathcal{F}$ with $|\mathcal{F}| = m$, and not having property $\mathcal{P}$.

([i]Dan Schwarz[/i])