Frequent elements in union-closed set families

TL;DR

This paper proves a generalized lower bound on the frequency of the $k$th-most popular element in union-closed families, extending the Union-Closed Sets Conjecture and characterizing extremal cases.

Contribution

It combines entropic and combinatorial methods to establish bounds for all $k \, \ge \, 2$ and characterizes families achieving equality.

Findings

Proves the $k$th-most popular element appears in at least $\frac{1}{2^{k-1}+1}$ of the sets.

Shows that as $|\mathcal{F}| \to \infty$, the $k$th-most frequent element appears in at least $(\frac{3 - \sqrt{5}}{2} - o(1))|\mathcal{F}|$ sets.

Characterizes families that achieve the equality in the bounds.

Abstract

The Union-Closed Sets Conjecture asks whether every union-closed set family $\mathcal{F}$ has an element contained in half of its sets. In 2022, Nagel posed a generalisation of this problem, suggesting that the $k$th-most popular element in a union-closed set family must be contained in at least $\frac{1}{2^{k-1} + 1} |\mathcal{F}|$ sets. We combine the entropic method of Gilmer with the combinatorial arguments of Knill to show that this is indeed the case for all $k \ge 2$, and characterise the families that achieve equality. Furthermore, we show that when $|\mathcal{F}| \to \infty$, the $k$th-most frequent element will appear in at least $\left( \frac{3 - \sqrt{5}}{2} - o(1) \right) |\mathcal{F}|$ sets, reflecting the recent progress made for the Union-Closed Set Conjecture.