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 $\mathcal{F}$?



What is the maximum size of a set system where the intersection of any two incomparable members is not in the set?
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>. 


