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

TL;DR

This paper extends Frankl's recent results on the size of differences in intersecting families of sets to the broader class of t-intersecting families, providing new bounds and insights.

Contribution

It generalizes Frankl's upper bound on the difference collection size from intersecting to t-intersecting families of sets.

Findings

Extended Frankl's bound to t-intersecting families.

Provided new upper bounds on difference collections.

Enhanced understanding of structure 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 called a $t$-intersecting family, if for some positive integer $t$ and any two members $F, G \in \mathcal{F}$ we have $|F\cap G| \geq t$. 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.