Extremal Systems with Upper-Bounded Odd Intersections

In this paper we determine the maximum cardinality m of a family A = {A1, ..., Am} of subsets of a set of n elements with the property that for each Ai there are at most s Aj such that |Ai∩Aj | is odd (even). A tight upper bound is given in the case s < c 2n/2 n . In the remaining cases we give an asymptotically tight upper bound. As an application we give a tight upper-bound for the cardinality of a family with even multi-intersection. Both results generalize a result of Berlekamp and Graver.