Best possible bounds on the number of distinct differences in intersecting families

Abstract

For a family $\mathcal{F}$, let $\mathcal{D}(\mathcal{F})$ stand for the family of all sets that can be expressed as $F\setminus G$, where $F,G\in \mathcal{F}$. A family $\mathcal{F}$ is intersecting if any two sets from the family have non-empty intersection. In this paper, we study the following question: what is the maximum of $|\mathcal{D}(\mathcal{F})|$ for an intersecting family of $k$-element sets? Frankl conjectured that the maximum is attained when $\mathcal{F}$ is the family of all sets containing a fixed element. We show that this holds if $n\ge 50k\ln k$ and $k\ge 50$. At the same time, we provide a counterexample for $n<4k$.