A note on distinct differences in $t$-intersecting families

Abstract

For a family $\mathcal{F}$ of subsets of $\{1,2,\dots ,n\}$, let $\mathcal{D}(\mathcal{F})=\{F\setminus G:F,G\in \mathcal{F}\}$ be the collection of all (setwise) differences of $\mathcal{F}$. The family $\mathcal{F}$ is called a $t$-intersecting family, if for some positive integer $t$ and any two members $F,G\in \mathcal{F}$ we have $|F\cap G|\ge t$. The family $\mathcal{F}$ is simply called intersecting if $t=1$. Recently, Frankl proved an upper bound on the size of $\mathcal{D}(\mathcal{F})$ for the intersecting families $\mathcal{F}$. In this note we extend the result of Frankl to $t$-intersecting families.