Frequent elements in union-closed set families

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.

12 pages Simplified and strengthened our proofs to obtain Nagel's conjecture in full