  The problem of bounding the size of a set system under various intersection
restrictions has a central place in extremal combinatorics. We investigate the
maximum number of disjoint pairs a set system can have in this setting. In
particular, we show that for any pair of set systems $(\mathcal{A},
\mathcal{B})$ which avoid a cross-intersection of size $t$, the number of
disjoint pairs $(A, B)$ with $A \in \mathcal{A}$ and $B \in \mathcal{B}$ is at
most $\sum_{k=0}^{t-1}\binom{n}{k}2^{n-k}$. This implies an asymptotically best
possible upper bound on the number of disjoint pairs in a single $t$-avoiding
family $\mathcal{F} \subset \mathcal{P}[n]$. We also study this problem when
$\mathcal{A}$, $\mathcal{B} \subset [n]^{(r)}$ are both $r$-uniform, and show
that it is closely related to the problem of determining the maximum of the
product $|\mathcal{A}||\mathcal{B}|$ when $\mathcal{A}$ and $\mathcal{B}$ avoid
a cross-intersection of size $t$, and $n \ge n_0(r, t)$.
