On the Beck-Fiala Conjecture for Random Set Systems

Abstract

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems $(X,\Sigma )$, where each element $x\in X$ lies in $t$ randomly selected sets of $\Sigma$, where $t$ is an integer parameter. We provide new bounds in two regimes of parameters. We show that when $|\Sigma |\ge |X|$ the hereditary discrepancy of $(X,\Sigma )$ is with high probability $O(\sqrt{t\log t})$; and when $|X|\gg |\Sigma {|}^t$ the hereditary discrepancy of $(X,\Sigma )$ is with high probability $O(1)$. The first bound combines the Lov{á}sz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.