On the Beck-Fiala Conjecture for Random Set Systems

TL;DR

This paper investigates discrepancy bounds for random sparse set systems motivated by the Beck-Fiala conjecture, providing new probabilistic bounds in different parameter regimes using combinatorial and probabilistic techniques.

Contribution

It introduces novel bounds on hereditary discrepancy for random set systems in two regimes, employing the Lovász Local Lemma and lattice analysis.

Findings

Hereditary discrepancy is O(√(t log t)) when |Σ| ≥ |X|

Hereditary discrepancy is O(1) when |X| ≫ |Σ|^t

New bounds improve understanding of discrepancy in 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{\'a}sz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.