Would it be reasonable to conjecture what follows : there is a real constant $c > 1/2$ such that, for every natural number $n$, if $X_{1}, \ldots , X_{n}$ is a union-stable family of distinct finite sets with at least two elements in their union $U$, then there is at least one subset of $U$ with two elements that intersects at least $cn - 1$ sets $X_{i}$ ? Thanks in advance.

As Miroslav Chlebik shows it here :

https://gowers.wordpress.com/2016/02/13/func3-further-strengthenings-and-variants/#comment-154441

the answer is yes with c = 3/4 if Frankl's conjecture is right.

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