Strong stability of 3-wise $t$-intersecting 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 $G\in{\mathcal G}$. For example, if ${\mathcal G}$ consists of all subsets containing a fixed $t$-element set, then it is a $3$-wise $t$-intersecting family with the measure $p^t$. Let $0<p\leq 2/(\sqrt{4t+9}-1)$, $\delta>0$, and let ${\mathcal G}$ be a $3$-wise $t$-intersecting family. It is known that the measure of ${\mathcal G}$ is at most $p^t$. Suppose, moreover, that ${\mathcal G}$ has the measure at least $(\frac12+\delta)p^t$. We show that, by choosing $t$ sufficiently large depending on $\delta$, the structure of ${\mathcal G}$ is one of (i) and (ii): (i) every subset in ${\mathcal G}$ contains a fixed $t$-element set, (ii) every subset in ${\mathcal G}$ contains at least $t+2$ elements from a fixed $(t+3)$-element set.