On the union of intersecting families

David Ellis, Noam Lifshitz

Published 2016-10-10Version 1

We prove that for any integer $r \geq 2$, if $X$ is an $n$-element set, and $\mathcal{F} = \mathcal{F}_1 \cup \mathcal{F}_2 \cup \ldots \cup \mathcal{F}_r$, where each $\mathcal{F}_i$ is an intersecting family of $k$-element subsets of $X$, then $|\mathcal{F}| \leq {n \choose k} - {n-r \choose k}$, provided $n \geq 2k+Ck^{2/3}$, where $C$ is a constant depending upon $r$ alone. Equality holds if and only if $\mathcal{F} = \{S \subset X:\ |S|=k,\ S \cap R \neq \emptyset\}$ for some $R \subset X$ with $|R|=r$. In the case $r=2$, this improves a result of Frankl and F\"uredi.