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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.

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    $\begingroup$ While this is a reasonable (but not great) question for MathOverflow, you might go to gowers.wordpress.com where some Polymath discussion covers questions similar to (and maybe exactly) this. If this question isn't there already, add it. Gerhard "Go In,The Water's Fine" Paseman, 2016.02.21. $\endgroup$ – Gerhard Paseman Feb 21 '16 at 20:01
  • $\begingroup$ Also, it is known that c=1/2 works when U is one of the X_i. You might be able to bring c down to exactly 1/2 by looking at those cases where U is a member of the family. Gerhard "Maybe Do The Hottub First" Paseman, 2016.02.21. $\endgroup$ – Gerhard Paseman Feb 21 '16 at 20:06
  • $\begingroup$ I don't understand. It is not interesting to bring c down to exactly 1/2, a value c > 1/2 is better than 1/2. And how could it be possible that U was not a member of the family ? (By the way, it could be interesting to look at the hypothetical value c = 2/3.) $\endgroup$ – Panurge Feb 22 '16 at 6:29
  • $\begingroup$ Sorry, I misread U as being a two element subset of the universe. My contention ( and I may be wrong ) is that there will be for every epsilon a union closed family with a two element set V as part of the family such that V intersects (1/2 + epsilon)n - 1 sets of the family and every other two element set intersects in strictly fewer sets of the family. So I think c= 2/3 will not work for all families. Gerhard " Union Closed Sets Are Tricky" Paseman, 2016.02.21. $\endgroup$ – Gerhard Paseman Feb 22 '16 at 6:59
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    $\begingroup$ Well, if you speak about this family with 17 sets indicated by Alec Edgington, the set {0,1} intersects 13 sets, which is in accordance with the value c = 2/3. $\endgroup$ – Panurge Feb 22 '16 at 10:33
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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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