Union-Closed Families with Small Average Overlap Densities

Abstract

In this very short paper, we show that the average overlap density of a union-closed family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ may be as small as \[\Theta((\log_2 \log_2 |\mathcal{F}|)/(\log_2 |\mathcal{F}|)),\] for infinitely many positive integers $n$.