Two Results on Union-Closed Families

TL;DR

This paper proves two new results about union-closed families: a density condition ensuring an element appears in at least half the sets, and a bound on the number of covering sets outside the family.

Contribution

It establishes a constant-density threshold for element frequency and bounds the number of external covering sets, advancing understanding of union-closed families.

Findings

If the family size exceeds a specific threshold, some element appears in at least half the sets.

The number of external sets covering the family is at most half of all possible sets.

Examples show the bounds are tight.

Abstract

We show that there is some absolute constant $c>0$, such that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}$, if \mbox{$|\mathcal{F}| \geq (\frac{1}{2}-c)2^n$}, then there is some element $i \in [n]$ that appears in at least half of the sets of $\mathcal{F}$. We also show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}$, the number of sets which are not in $\mathcal{F}$ that cover a set in $\mathcal{F}$ is at most $2^{n-1}$, and provide examples where the inequality is tight.