Affiliation:
1. School 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
Abstract
The union-closed sets conjecture states that, in any nonempty union-closed 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 information-theoretic 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 union-closed 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.
Funder
NSFC
Fundamental Research Funds for the Central Universities of China
Subject
General Physics and Astronomy
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