Constructing Large Set Systems with Given Intersection Sizes Modulo Composite Numbers

Abstract

We consider k-uniform set systems over a universe of size n such that the size of each pairwise intersection of sets lies in one of s residue classes mod q, but k does not lie in any of these s classes. A celebrated theorem of Frankl and Wilson [8] states that any such set system has size at most (ⁿ_s) when q is prime. In a remarkable recent paper, Grolmusz [9] constructed set systems of superpolynomial size Ω(exp(c log²n/log log n)) when q = 6. We give a new, simpler construction achieving a slightly improved bound. Our construction combines a technique of Frankl [6] of ‘applying polynomials to set systems’ with Grolmusz's idea of employing polynomials introduced by Barrington, Beigel and Rudich [5]. We also extend Frankl's original argument to arbitrary prime-power moduli: for any ε > 0, we construct systems of size n^s+g(s), where g(s) = Ω(s^1−ε). Our work overlaps with a very recent technical report by Grolmusz [10].