Constructing set systems with prescribed intersection sizes

Abstract

Let f be an n-variable polynomial with positive integer coefficients, and let H={H 1,H 2,…,H m} be a set system on the n-element universe. We define a set system f( H)={G 1,G 2,…,G m} and prove that f( H i 1 ∩ H i 2 ∩⋯∩ H i k )=| G i 1 ∩ G i 2 ∩⋯∩ G i k |, for any 1⩽ k⩽ m, where f( H i 1 ∩ H i 2 ∩⋯∩ H i k ) denotes the value of f on the characteristic vector of H i 1 ∩ H i 2 ∩⋯∩ H i k . 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.