Pairwise Intersection Profiles of Intersecting Families

Abstract

Let $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ be a $k$-uniform family, and let $P_m(\mathcal{F})$ denote the number of ordered pairs $(A,B)\in \mathcal{F}\times \mathcal{F}$ with $|A\cap B|=m$. We study the pairwise intersection profile $(P_0(\mathcal{F}),P_1(\mathcal{F}),\dots ,P_k(\mathcal{F}))$ of intersecting families. We prove that for every $1\le m\le k$, if $n\ge \max \{2k,3k-m-1\}$, then $$P_m(\mathcal{F})\le P_m({\mathcal{S}}_1^k),$$ where ${\mathcal{S}}_1^k$ is a full star. Moreover, if $n\ge \max \{2k+1,3k-m\}$, then equality holds if and only if $\mathcal{F}$ is a star up to permutations. As a corollary, for any function $f:\{0,1,\dots ,k\}\to {\mathbb{R}}_{\ge 0}$ and $n\ge 3k-2$, we show that ${\Phi }_f(\mathcal{F})={\sum }_{(A,B)\in \mathcal{F}\times \mathcal{F}}f(|A\cap B|)$ is maximized by a star. We also establish a stability result showing that, for $n\ge \max \{2k+1,3k-m\}$, if $P_m(\mathcal{F})$ is sufficiently close to its maximum, then $\mathcal{F}$ has an element of degree close to $\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)$.