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

TL;DR

This paper investigates the maximum number of distinct differences in intersecting families of sets, proving bounds under certain conditions and providing counterexamples for smaller cases.

Contribution

It establishes bounds on the maximum size of difference families in intersecting set families, confirming a conjecture for large parameters and offering counterexamples for smaller ones.

Findings

Maximum difference family size is achieved by all sets containing a fixed element when n ≥ 50k ln k and k ≥ 50.

Counterexamples exist for n < 4k, disproving the conjecture in that range.

The paper advances understanding of the structure of intersecting families and their difference sets.

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 $n< 4k$.