The maximum measure of non-trivial 3-wise intersecting families

Norihide Tokushige

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arXiv · 2203.17158

The maximum measure of non-trivial 3-wise intersecting families

Norihide Tokushige

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Abstract

Let $\mathcal G$ be a family of subsets of an $n$-element set. The family $\mathcal G$ is called non-trivial $3$-wise intersecting if the intersection of any three subsets in $\mathcal G$ is non-empty, but the intersection of all subsets is empty. For a real number $p\in(0,1)$ we define the measure of the family by the sum of $p^{|G|}(1-p)^{n-|G|}$ over all $G\in\mathcal G$. We determine the maximum measure of non-trivial $3$-wise intersecting families. We also discuss the uniqueness and stability of the corresponding optimal structure. These results are obtained by solving linear programming problems.

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Norihide Tokushige. 2022-03-31. The maximum measure of non-trivial 3-wise intersecting families. https://arxiv.org/abs/2203.17158

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