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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Veröffentlicht in:	The Electronic journal of combinatorics 2024-09, Vol.31 (3)
Hauptverfasser:	Alweiss, Ryan, Huang, Brice, Sellke, Mark
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Sprache:	eng
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container_title	The Electronic journal of combinatorics
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creator	Alweiss, Ryan Huang, Brice Sellke, Mark
description	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.
doi_str_mv	10.37236/12232
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title	Improved Lower Bound for Frankl’s Union-Closed Sets Conjecture
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