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TR98-044 | 31st July 1998 00:00
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#### Bounds on Pairs of Families with Restricted Intersections

**Abstract:**

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 $L=\{0,\ldots,k\}$.