TITLE: What is the maximum size of a set system where the intersection of any two incomparable members is not in the set?
QUESTION [12 upvotes]: Let the set $\mathcal{F}$ consist of subsets of $[n]$.  Suppose that for any incomparable $A$ and $B$ in $\mathcal{F}$, we have $A \cap B \notin \mathcal{F}$.  What is the largest possible size of $\mathcal{F}$?

REPLY [4 votes]: It was proved by Kleitman that $|\mathcal F|\le {n\choose n/2}+2^n/n$, see http://www.sciencedirect.com/science/article/pii/0097316576900376.