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arXiv · 2110.12708

Maximal $3$-wise intersecting families

József Balogh·Ce Chen·Kevin Hendrey·Ben Lund·Haoran Luo·Casey Tompkins·Tuan Tran
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Abstract

A family $\mathcal{F}$ on ground set $[n]:=\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of at most $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while maintaining this property. In 1974, Erd\H{o}s and Kleitman asked for the minimum size of a maximal $k$-wise intersecting family. We answer their question for $k=3$ and sufficiently large $n$. We show that the unique minimum family is obtained by partitioning the ground set $[n]$ into two sets $A$ and $B$ with almost equal sizes and taking the family consisting of all the proper supersets of $A$ and of $B$.

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József Balogh, Ce Chen, Kevin Hendrey, Ben Lund, Haoran Luo, Casey Tompkins, Tuan Tran. 2021-10-25. Maximal $3$-wise intersecting families. https://arxiv.org/abs/2110.12708

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