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.



number of sets including the most popular elements in intersecting sets family
This is completely false. Consider the lines of a finite projective plane. 


