Extension of a Method of Gilmer

TL;DR

This paper extends Gilmer's method to precisely determine the maximum bound for the element frequency in union-closed set families, achieving the conjectured optimal bound of approximately 38.1%.

Contribution

It finds the exact optimal value for Gilmer's bound using entropy and probabilistic analysis, improving previous constant bounds.

Findings

Achieves the conjectured optimal bound of ~38.1%.

Refines Gilmer's method with entropy-based analysis.

Provides a precise mathematical framework for the bound.

Abstract

It is a well-known conjecture, sometimes attributed to Frankl, that for any family of sets which is closed under the union operation, there is some element which is contained in at least half of the sets. Gilmer was the first to prove a constant bound, showing that there is some element contained in at least 1\% of the sets. They state in their paper that the best possible bound achievable by the same method is $\frac{3-\sqrt5}2\approx 38.1\%$. This note achieves that bound by finding the optimum value, given a binary variable $X$ potentially depending on some other variable $S$ with a given expected value $E(X)$ and conditional entropy $H(X|S)$ of the conditional entropy of $H(X_1\cup X_2|S_1,S_2)$ for independent readings $X_1, S_1$ and $X_2,S_2$.