Intersecting families with 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.