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.
Categories: math.CO