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 $n< 4k$.