Dimension-Free Bounds for the Union-Closed Sets Conjecture

TL;DR

This paper advances the information-theoretic approach to the union-closed sets conjecture, deriving explicit, improved bounds on the element frequency proportion, including a numerically evaluated bound of approximately 0.38234.

Contribution

It extends Gilmer's technique to produce explicit bounds, improving previous results and making Sawin's improvement computationally accessible.

Findings

Derived new bounds for the union-closed sets conjecture

Numerically evaluated a bound of approximately 0.38234

Unified previous improvements into a single optimization framework

Abstract

The union-closed sets conjecture states that in any nonempty union-closed family $\mathcal{F}$ of subsets of a finite set, there exists an element contained in at least a proportion $1/2$ of the sets of $\mathcal{F}$. Using the information-theoretic method, Gilmer \cite{gilmer2022constant} recently showed that there exists an element contained in at least a proportion $0.01$ of the sets of such $\mathcal{F}$. He conjectured that his technique can be pushed to the constant $\frac{3-\sqrt{5}}{2}$ which was subsequently confirmed by several researchers \cite{sawin2022improved,chase2022approximate,alweiss2022improved,pebody2022extension}. Furthermore, Sawin \cite{sawin2022improved} showed that Gilmer's technique can be improved to obtain a bound better than $\frac{3-\sqrt{5}}{2}$, but this new bound is not explicitly given by Sawin. This paper further improves Gilmer's technique to derive new bounds in the optimization form for the union-closed sets conjecture. These bounds include Sawin's improvement as a special case. By providing cardinality bounds on auxiliary random variables, we make Sawin's improvement computable, and then evaluate it numerically which yields a bound around $0.38234$, slightly better than $\frac{3-\sqrt{5}}{2}\approx0.38197$. }