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 $\binom{ X}{ r}$ 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 $\binom{ [n]}{ a} \cup \binom{ [2n]}{ b}$ whenever $n > b$. For $n$ sufficiently large, we also determine the maximum size of an intersecting family in $\binom{ [2n]}{ a} \cup \binom{ [n+1, 3n]}{ a} \cup \binom{ [n] \cup [2n + 1, 3n]}{ a} \cup \binom{ [3n]}{ b}$ 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.
Categories: math.CO