A proof of the union-closed sets conjecture

TL;DR

This paper presents a proof of the long-standing union-closed sets conjecture by refining Gilmer's entropy approach, involving a convex combination of independent samples from the family.

Contribution

It introduces a novel refinement of the entropy method by considering convex combinations of independent samples, advancing the proof of the union-closed sets conjecture.

Findings

Proof of the union-closed sets conjecture established

Refinement of entropy-approach with convex combinations

New technique potentially applicable to related combinatorial problems

Abstract

We provide a proof of the union-closed sets conjecture, by means of a suitable refinement of the breakthrough entropy-approach introduced by Gilmer. The novelty here is to consider a convex combination of $A$ and $A\cup B$, where $A,B$ are independent samples from the uniform distribution over a union-closed family.