DimensionFree Bounds for the UnionClosed Sets Conjecture
Abstract
The unionclosed sets conjecture states that, in any nonempty unionclosed family F of subsets of a finite set, there exists an element contained in at least a proportion 1/2 of the sets of F. Using an informationtheoretic method, Gilmer recently showed that there exists an element contained in at least a proportion 0.01 of the sets of such F. He conjectured that their technique can be pushed to the constant 3‑52 which was subsequently confirmed by several researchers including Sawin. Furthermore, Sawin also showed that Gilmer's technique can be improved to obtain a bound better than 3‑52 but this new bound was not explicitly given by Sawin. This paper further improves Gilmer's technique to derive new bounds in the optimization form for the unionclosed 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 approximately 0.38234, slightly better than 3‑52≈0.38197.
 Publication:

Entropy
 Pub Date:
 May 2023
 DOI:
 10.3390/e25050767
 arXiv:
 arXiv:2212.00658
 Bibcode:
 2023Entrp..25..767Y
 Keywords:

 unionclosed sets conjecture;
 informationtheoretic method;
 coupling;
 Mathematics  Combinatorics;
 Computer Science  Information Theory
 EPrint:
 10 pages, to appear in Entropy as an invited paper