Improved Lower Bound for Frankl’s Union-Closed Sets Conjecture

We verify an explicit inequality conjectured in [Gilmer, 2022, arXiv:2211.09055], thus proving that for any nonempty union-closed family $\mathcal{F} \subseteq 2^{[n]}$, some $i\in [n]$ is contained in at least a $\frac{3-\sqrt{5}}{2} \approx 0.38$ fraction of the sets in $\mathcal{F} \$. One case,...

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Bibliographic Details
Published inThe Electronic journal of combinatorics Vol. 31; no. 3
Main Authors Alweiss, Ryan, Huang, Brice, Sellke, Mark
Format Journal Article
LanguageEnglish
Published 20.09.2024
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Summary:We verify an explicit inequality conjectured in [Gilmer, 2022, arXiv:2211.09055], thus proving that for any nonempty union-closed family $\mathcal{F} \subseteq 2^{[n]}$, some $i\in [n]$ is contained in at least a $\frac{3-\sqrt{5}}{2} \approx 0.38$ fraction of the sets in $\mathcal{F} \$. One case, an explicit one-variable inequality, is checked by computer calculation.
ISSN:1077-8926
1077-8926
DOI:10.37236/12232