Intersecting families with full difference sets

TL;DR

This paper investigates the maximum size of difference sets in intersecting families of subsets, extending known results to cases where the difference set is 'full' for certain parameters, and constructs examples for even k.

Contribution

It proves the existence of intersecting families with full difference sets for even k ≥ 4, answering an open question posed by Frankl.

Findings

Existence of intersecting families with full difference sets for even k ≥ 4.

Extension of previous bounds on n for maximum difference set size.

Construction of specific families achieving full difference sets.

Abstract

For a family $\mathcal{F}$ of subsets of a finite set, define $\mathcal{D}(\mathcal{F})=\{F\setminus F': F, F'\in\mathcal{F}\}$. A family $\mathcal{F}$ is called intersecting if $F\cap F'\not=\emptyset$ for all $F, F'\in\mathcal{F}$. Frankl \cite{Frankl} showed that for a $k$-uniform intersecting family $\mathcal{F}\subset{[n]\choose k}$ with $n\ge k(k+3)$, $|\mathcal{D}(\mathcal{F})|$ reaches the maximum if and only if $\mathcal{F}$ is a $k$-uniform full star. Later, Frankl-Kiselev-Kupavskii \cite{FKK} improved the bound $n\ge k(k+3)$ in the above result of Frankl \cite{Frankl} to $n\ge 50klnk$ for $k\ge 50$. For $2k<n<4k$, Frankl-Kiselev-Kupavskii \cite{FKK} showed that there exists a $k$-uniform family $\mathcal{F}\subset{[n]\choose k}$ such that $|\mathcal{D}(\mathcal{F})|$ is larger than $|\mathcal{D}(\mathcal{S})|$, where $\mathcal{S}$ is a full star. This result left the case $n=2k$ open and we show that $\mathcal{D}(\mathcal{F})$ can be `full' for some $\mathcal{F}\subset{[n]\choose k}$. It is clear that for an intersecting family $\mathcal{F}\subset{[n]\choose k}$, $\mathcal{D}(\mathcal{F})\subseteq \cup_{j=0}^{k-1}{[n]\choose j}$. We say that a $k$-uniform intersecting family $\mathcal{F}\subset{[n]\choose k}$ has full differences if $\mathcal{D}(\mathcal{F})=\cup_{j=0}^{k-1}{[n]\choose j}$. For odd $k$, Frankl \cite{Frankl} gave a $k$-uniform intersecting family $\mathcal{F}\subset{[2k]\choose k}$ having full differences of size $k-1$, and he asked for even $k\ge 4$ whether there exists a $k$-uniform intersecting family $\mathcal{F}\subset{[2k]\choose k}$ having full differences of size $k-1$. We answer this question in a stronger form and show that for even $k\ge 4$, there exists a $k$-uniform intersecting family $\mathcal{F}\subset{[2k]\choose k}$ having full differences.