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Dimension-Free Bounds for the Union-Closed Sets Conjecture.

Lei Yu1

  • 1School of Statistics and Data Science, The Key Laboratory of Pure Mathematics and Combinatorics (LPMC), Key Laboratory for Medical Data Analysis and Statistical Research of Tianjin (KLMDASR), and Laboratory for Economic Behaviors and Policy Simulation (LEBPS), Nankai University, Tianjin 300071, China.

Entropy (Basel, Switzerland)
|May 27, 2023
PubMed
Summary

Researchers improved bounds for the union-closed sets conjecture, a problem in combinatorics. New numerical results offer a slightly better bound than previously known, advancing this area of mathematical research.

Keywords:
couplinginformation-theoretic methodunion-closed sets conjecture

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Area of Science:

  • Combinatorics
  • Set Theory
  • Discrete Mathematics

Background:

  • The union-closed sets conjecture posits that any non-empty union-closed family of subsets of a finite set must contain an element present in at least half the subsets.
  • Previous work by Gilmer established a lower bound of 0.01 using information-theoretic methods, which was later improved to 3/52 by Sawin and others.
  • Sawin indicated a potential for further improvement beyond 3/52 using Gilmer's technique, but did not explicitly state the new bound.

Purpose of the Study:

  • To further enhance Gilmer's information-theoretic technique for the union-closed sets conjecture.
  • To derive new, improved bounds in an optimization form.
  • To make Sawin's previously unstated improvement computable and evaluate it numerically.

Main Methods:

  • Refinement of the information-theoretic method originally developed by Gilmer.
  • Derivation of new bounds expressed in an optimization framework.
  • Analysis of auxiliary random variables by providing cardinality bounds to enable computation.

Main Results:

  • The study presents improved bounds for the union-closed sets conjecture, encompassing Sawin's improvement as a specific instance.
  • Numerical evaluation of the refined technique yields a bound of approximately 0.38234.
  • This new bound is slightly better than the previously established bound of 3/52 (approximately 0.38197).

Conclusions:

  • The paper successfully improves upon existing information-theoretic bounds for the union-closed sets conjecture.
  • The derived numerical bound of ~0.38234 represents a marginal but significant advancement.
  • The methodology provides a computable path for further theoretical and numerical exploration of the conjecture.