Union-Closed Families with Small Average Overlap Densities

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 28 Jan 2022Embargo end date: 01 Jan 2020 United Kingdom Publisher:The Electronic Journal of CombinatoricsJournal:The Electronic Journal of Combinatorics, volume 29 (eissn: 1077-8926,

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Authors: Ellis, David;

doi: 10.37236/10121 , 10.48550/arxiv.2012.03235

arXiv: 2012.03235

handle: 1983/18d79451-82e6-4f39-b9ab-a7ab83ef6b01

Union-Closed Families with Small Average Overlap Densities

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Abstract

In this very short paper, we show that the average overlap density of a union-closed family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ may be as small as \[\Theta((\log_2 \log_2 |\mathcal{F}|)/(\log_2 |\mathcal{F}|)),\] for infinitely many positive integers $n$.

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Keywords

05D05, 330, union-closed conjecture, Extremal set theory, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), average overlap density, 510

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