Three-sets in a Union-Closed Family

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 \( \frac{2n}{3} \).