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

TL;DR

This paper determines the maximum measure of non-trivial 3-wise intersecting families of subsets, providing insights into their structure and stability, using linear programming techniques.

Contribution

It establishes the maximum measure for such families and analyzes their uniqueness and stability, advancing understanding of intersecting set systems.

Findings

Maximum measure of non-trivial 3-wise intersecting families identified

Uniqueness and stability of optimal structures discussed

Linear programming used to derive results

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.