摘要：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| >= |X| the hereditary discrepancy of (X,Sigma) is with high probability O(sqrt{t log t}); and when |X| >> |\Sigma|^t the hereditary discrepancy of (X,Sigma) is with high probability O(1). The first bound combines the Lovasz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.
关键词：Discrepancy theory; Beck-Fiala conjecture; Random set systems
