Union-closed families with small average overlap densities

TL;DR

This paper demonstrates that the average overlap density in union-closed families can be extremely small, decreasing roughly as the ratio of double logarithm to logarithm of the family size, for infinitely many cases.

Contribution

It establishes a new bound on the minimal average overlap density in union-closed families, revealing it can be significantly smaller than previously understood.

Findings

Average overlap density can be as small as Θ((log log |F|)/(log |F|))

This bound holds for infinitely many positive integers n

Highlights the potential for very low overlap densities in union-closed families

Abstract

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