  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.
