Three-sets in a Union-Closed Family

Abstract

A union closed (UC) family $\mathcal{A}$ is a finite family of sets such that the union of any two sets in $\mathcal{A}$ is also in $\mathcal{A}$. Peter Frankl conjectured in 1979 that for every union closed family $\mathcal{A}$, there exists some $x$ contained in at least half the members of $\mathcal{A}$. In this paper, we show that if a UC family $\mathcal{A}$ fails the conjecture, then no element can appear in more than two of its $3$-sets, and so the number of $3$-sets in $\mathcal{A}$ can be no more than $\frac{2n}{3}$.