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

Disjoint pairs in set systems with restricted intersection