On the maximum number of distinct intersections in an intersecting family

Abstract

For $n>2k\ge 4$ we consider intersecting families $\mathcal{F}$ consisting of $k$-subsets of $\{1,2,\dots ,n\}$. Let $\mathcal{I}(\mathcal{F})$ denote the family of all distinct intersections $F\cap F^{\prime }$, $F\ne F^{\prime }$ and $F,F^{\prime }\in \mathcal{F}$. Let $\mathcal{A}$ consist of the $k$-sets $A$ satisfying $|A\cap \{1,2,3\}|\ge 2$. We prove that for $n\ge 50k^2$ $|\mathcal{I}(\mathcal{F})|$ is maximized by $\mathcal{A}$.