Intersecting non-uniform families containing subfamilies

Abstract

A family of sets is said to be intersecting if every pair of sets in the family have non-empty intersection. In this paper, we initiate the study of intersecting non-uniform families of sets of one of two sizes containing given subfamilies. For a set $X$ and integer $r$, let $\left(\genfrac{}{}{0ex}{}{X}{r}\right)$ denote the family $\{A\subseteq X:|X|=r\}$. Let $a$, $b$, and $n$ be positive integers such that $a<b$. We determine the maximum size of an intersecting family in $\left(\genfrac{}{}{0ex}{}{[n]}{a}\right)\cup \left(\genfrac{}{}{0ex}{}{[2n]}{b}\right)$ whenever $n>b$. For $n$ sufficiently large, we also determine the maximum size of an intersecting family in $\left(\genfrac{}{}{0ex}{}{[2n]}{a}\right)\cup \left(\genfrac{}{}{0ex}{}{[n+1,3n]}{a}\right)\cup \left(\genfrac{}{}{0ex}{}{[n]\cup [2n+1,3n]}{a}\right)\cup \left(\genfrac{}{}{0ex}{}{[3n]}{b}\right)$ whenever $3n>2b$ and $b>a+2$. Our results are, in some sense, best possible. Our methods include the use of Katona's shadow intersection theorem and a recent diversity theorem of Kupavskii and~Zakharov.

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• Structure and properties of large intersecting families