Extension of a Method of Gilmer
Abstract
It is a wellknown conjecture, sometimes attributed to Frankl, that for any family of sets which is closed under the union operation, there is some element which is contained in at least half of the sets. Gilmer was the first to prove a constant bound, showing that there is some element contained in at least 1\% of the sets. They state in their paper that the best possible bound achievable by the same method is $\frac{3\sqrt5}2\approx 38.1\%$. This note achieves that bound by finding the optimum value, given a binary variable $X$ potentially depending on some other variable $S$ with a given expected value $E(X)$ and conditional entropy $H(XS)$ of the conditional entropy of $H(X_1\cup X_2S_1,S_2)$ for independent readings $X_1, S_1$ and $X_2,S_2$.
 Publication:

arXiv eprints
 Pub Date:
 November 2022
 DOI:
 10.48550/arXiv.2211.13139
 arXiv:
 arXiv:2211.13139
 Bibcode:
 2022arXiv221113139P
 Keywords:

 Mathematics  Combinatorics