On set systems with restricted intersections modulo a composite number

Abstract

Let S be a set of n elements, and let H be a set-system on S, which satisfies that the size of any element of H is divisible by m but the intersection of any two elements of H is not divisible by m If m is a prime or prime-power, then the famous Frankl-Wilson theorem [3] implies that |H| = O(n ^m-1), i.e. for fixed m, its size is at most polynomial in n. This theorem has numerous applications in combinatorics and also in geometry, (c.f. the disproof of Borsuk's conjecture by Kahn and Kalai in 1992 ([4]), or explicit constructions of Ramsey graphs, or other geometric applications related to the Hadwiger-problem.) Frankl and Wilson asked in [3] whether an analogous upper bound existed for non-prime power, composite moduli. Here we show a surprising construction of a superpolynomial-sized uniform set-system H satisfying the intersection-property, for every non-prime-power, composite m, negatively settling a related conjecture of Babai and Frankl [1]. The proof uses a low-degree polynomial-construction of Barrington, Beigel and Rudich [2], and a new method (Lemma 8), for constructing set-systems from multivariate polynomials.