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\}.