We study pairs of families ${\cal A},{\cal B}\subseteq
2^{\{1,\ldots,r\}}$ such that $|A\cap B|\in L$ for any
$A\in{\cal A}$, $B\in{\cal B}$. We are interested in the maximal
product $|{\cal A}|\cdot|{\cal B}|$, given $r$ and $L$. We give
asymptotically optimal bounds for $L$ containing only elements
of $s<q$ residue classes modulo $q$, where $q$ is arbitrary
(even non-prime) and $s$ is a constant. As a consequence, we
obtain a version of Frankl-R\"{o}dl result about forbidden
intersections for the case of two forbidden intersections. We
also give tight bounds for $L=\{0,\ldots,k\}$.