Constructing set systems with prescribed intersection sizes

Let f be an n-variable polynomial with positive integer coefficients, and let H = {H1, H2,..., Hm} be a set system on the n-element universe. We define a set system f(H) = {G1, G2,..., Gm} and prove that f(Hi1 ∩ Hi2 ∩... ∩ Hik) = |Gi1 ∩ Gi2 ∩ ... ∩ Gik|, for any 1 ≤ k ≤ m, where f(Hi1 ∩ Hi2 ∩... ∩Hik) denotes the value of f on the characteristic vector of Hi1 ∩ Hi2 ∩... ∩ Hik. The construction of f(H) is a straightforward polynomial-time algorithm from the set system H and the polynomial f. In this paper we use this algorithm for constructing set systems with prescribed intersection sizes modulo an integer. As a by-product of our method, some upper bounds on the number of sets in set systems with prescribed intersection sizes are extended.

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