The $11$-element case of Frankl's conjecture

Abstract

In 1979, P. Frankl conjectured that in a finite union-closed family ${\cal F}$ of finite sets, ${\cal F}\neq\{\emptyset\}$, there has to be an element that belongs to at least half of the sets in ${\cal F}$. We prove this when $|\bigcup{\cal F}|\leq 11$.