Strong stability of 3-wise $t$-intersecting families
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 $p^t$. For a given $\delta>0$, by choosing $t$ sufficiently large, the following holds for all $p$ with $0<p\leq 2/(\sqrt{4t+9}-1)$. If $\mathcal G$ is a $3$-wise $t$-intersecting family with the measure at least $(\frac12+\delta)p^t$, then $\mathcal G$ satisfies 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.