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Better bounds for the union-closed sets conjecture using the entropy approach

Stijn Cambie

TL;DR

The paper advances the entropy-based approach to the union-closed conjecture by solving Sawin's optimization question exactly, enabling an improved bound on the fraction of sets containing a given element. The authors develop a framework that uses dependent, element-wise sampling and reduces the resulting optimization to low-dimensional probability distributions, with computer-assisted verification guiding the final steps. The main result establishes a sharper constant $c \approx 0.3823455$, demonstrating that in any nonempty union-closed family $\mathcal{F}$, some element lies in at least $c|\mathcal{F}|$ sets. This refines the understanding of the entropy method's limits and provides a concrete pathway toward potential eventual resolution of the conjecture, while highlighting the necessity of combining sampling dependencies with careful distributional analysis. The work also clarifies the boundaries of Gilmer's entropy approach and offers a replicable computational approach for verifying related optimization problems in extremal combinatorics.

Abstract

We improve the best known constant $\frac{3-\sqrt 5}{2}$ for which the union-closed conjecture is known to be true, by using dependent samples as suggested by Sawin and the entropy approach on this problem initiated by Gilmer. Meanwhile, we focus on the intuition behind this entropy approach and its boundaries.

Better bounds for the union-closed sets conjecture using the entropy approach

TL;DR

The paper advances the entropy-based approach to the union-closed conjecture by solving Sawin's optimization question exactly, enabling an improved bound on the fraction of sets containing a given element. The authors develop a framework that uses dependent, element-wise sampling and reduces the resulting optimization to low-dimensional probability distributions, with computer-assisted verification guiding the final steps. The main result establishes a sharper constant , demonstrating that in any nonempty union-closed family , some element lies in at least sets. This refines the understanding of the entropy method's limits and provides a concrete pathway toward potential eventual resolution of the conjecture, while highlighting the necessity of combining sampling dependencies with careful distributional analysis. The work also clarifies the boundaries of Gilmer's entropy approach and offers a replicable computational approach for verifying related optimization problems in extremal combinatorics.

Abstract

We improve the best known constant for which the union-closed conjecture is known to be true, by using dependent samples as suggested by Sawin and the entropy approach on this problem initiated by Gilmer. Meanwhile, we focus on the intuition behind this entropy approach and its boundaries.
Paper Structure (14 sections, 5 theorems, 20 equations, 1 figure)

This paper contains 14 sections, 5 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Terminology and notation
  3. Preliminaries
  4. Why the entropy approach works for the union-closed conjecture
  5. Observations on approximate union-closed set system
  6. Upper bound for Question \ref{['ques:Sawin']} and limits on the approach of Sawin
  7. Summary of the proof
  8. Proof for the optimal constant of Question \ref{['ques:Sawin']}
  9. Reduction of support of the probability distribution
  10. Reduction to small support of the probability distribution
  11. Verification for distributions with support of size at most $3$
  12. Precise verification
  13. Proof of bound for sharper union-closed conjecture
  14. Open access statement.

Key Result

Theorem 3

Let $\mathcal{F} \subset 2^{[n]}$ be a nonempty union-closed family. Then there is some element $i \in [n]$ that appears in at least $c\lvert \mathcal{F}\rvert$ many sets of $\mathcal{F}.$

Figures (1)

  • Figure 1: An example of improving the distribution

Theorems & Definitions (13)

  • Conjecture 1: Union-closed conjecture
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Claim 7
  • proof : Proof
  • ...and 3 more