# number of sets including the most popular elements in intersecting sets family

Let $$F$$ be a set consisting of some subsets of $$[n]$$, and any two sets in $$F$$ have at least one element in common. I think I read a result stating as following: there exists an element $$x$$, such that at least half of $$F$$ include $$x$$. But I do not remember the related reference. Do anyone know the reference or some similar results?Thanks.

• Frankl union-closed sets conjecture? – Seva Feb 12 at 12:25
• You may be thinking of en.wikipedia.org/wiki/Union-closed_sets_conjecture – bof Feb 12 at 12:28
• In view of the given link to the Frankl conjecture and Fedor's answer, it seems that the missing assumption in your statement is that $F$ should be closed under taking unions. – YCor Feb 12 at 17:00

• Sorry for not stating the problem clearly. $F$ is consisting of subsets of $[n]$ – xmchenhit Feb 12 at 12:32
• So what? Enumerate the points from 1 to $n$. – Fedor Petrov Feb 12 at 12:44
Is such generality the result is false. If you have $$n$$ sets, you can always define $${n \choose 2}$$ elements such that each of them is contained in a different couple of sets. This way all the sets have non-empty intersection and no element is contained in more than $$2$$ sets.