Improved Lower Bound for Frankl's Union-Closed Sets Conjecture

TL;DR

This paper proves a new lower bound for the frequency of an element in any nonempty union-closed family, confirming a recent conjecture and using computational verification for part of the proof.

Contribution

It establishes an explicit inequality confirming a conjecture, improving the known lower bound for element frequency in union-closed families.

Findings

Some element appears in at least 38% of the sets

The conjecture by Gilmer is verified for the given bound

Computer-aided proof for a specific inequality

Abstract

We verify an explicit inequality conjectured recently by Gilmer, thus proving that for any nonempty union-closed family $F \subseteq 2^{[n]}$, some $i\in [n]$ is contained in at least a $\frac{3-\sqrt{5}}{2} \approx 0.38$ fraction of the sets in $F$. One case, an explicit one-variable inequality, is checked by computer calculation.