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

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.

arXiv comment30 pages