Consider a union-closed family $\mathcal{F}=\{A_1,…,A_n\}$ of $n \gt 1$ finite sets.
I was not able to find a counterexample to the following conjecture:
there exist two sets $A,B \in \mathcal{F}$ such that $A \setminus B \not= \emptyset$ is a (not necessarily proper) subset of at least half of the sets of $\mathcal{F}$.
Obviously it implies the union-closed sets conjecture.
Is it possible to find a counterexample? Or if not, is the above conjecture implied by the union-closed sets conjecture?
