The maximum measure of 3-wise t-intersecting families

TL;DR

This paper determines the maximum measure of 3-wise t-intersecting families of subsets for large t and specific p, establishing sharp bounds and stability results for such families.

Contribution

It provides a precise maximum measure for 3-wise t-intersecting families when t≥15 and p is below a certain threshold, including sharpness and stability analysis.

Findings

Maximum measure is p^t for given conditions.

Bound on p is sharp.

Stability results for shifted families.

Abstract

Let $\mathcal G$ be a family of subsets of an $n$-element set. The family $\mathcal G$ is called $3$-wise $t$-intersecting if the intersection of any three subsets in $\mathcal G$ is of size at least $t$. 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 prove that if $t\geq 15$ and $p\leq 2/(\sqrt{4t+9}-1)$ then $p^t$ is the maximum measure of $3$-wise $t$-intersecting families, and the bound for $p$ is sharp. We also present the corresponding stability result for shifted families.