A cross-intersection theorem for subsets of a set

Abstract: Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq k}$ denote the family of all subsets of ${1, \dots, n}$ of size at most $k$. We show that if $\mathcal{A} \subseteq {[m] \choose \leq r}$, $\mathcal{B} \subseteq {[n] \choose \leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then [|\mathcal{A}||\mathcal{B}| \leq \sum_{i=0}^r {m-1 \choose i-1} \sum_{j=0}^s {n-1 \choose j-1},] and equality holds if $\mathcal{A} = {A \in {[m] \choose \leq r} \colon 1 \in A}$ and $\mathcal{B} = {B \in {[n] \choose \leq s} \colon 1 \in B}$. Also, we generalise this to any number of such cross-intersecting families.