Bounds on Pairs of Families with Restricted Intersections

We study pairs of families A; B 2 f1;:::;rg such that jA \ Bj 2 L for any A 2 A, B 2 B. We are interested in the maximum product jAj jBj, 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-RR odl result about forbidden intersections for the case of two forbidden intersections. We also give tight bounds for L = f0; : : : ; kg.