The union-closed sets conjecture for non-uniform distributions
Abstract
The union-closed sets conjecture, attributed to P\'eter Frankl from 1979, states that for any non-empty finite union-closed family of finite sets not consisting of only the empty set, there is an element that is in at least half of the sets in the family. We prove a version of Frankl's conjecture for families distributed according to any one of infinitely many distributions. As a corollary, in the intersection-closed reformulation of Frankl's conjecture, we obtain that it is true for families distributed according to any one of infinitely many Maxwell--Boltzmann distributions with inverse temperatures bounded below by a positive universal constant. Frankl's original conjecture corresponds to zero inverse temperature.
Cite
@article{arxiv.2305.19338,
title = {The union-closed sets conjecture for non-uniform distributions},
author = {Masoud Zargar},
journal= {arXiv preprint arXiv:2305.19338},
year = {2023}
}
Comments
15 pages. Exposition slightly improved, a couple minor typos fixed, and a reference updated. Comments welcome