A Note on Distinct Differences in t-Intersecting Families

Abstract

For a family $\mathcal {F}$ of subsets of $\{1,2,\ldots ,n\}$, 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 said to be a t-intersecting family for some positive integer t if $|F\cap G| \ge t$ for all $F, G \in \mathcal {F}$. 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.