On the discrepancy of random low degree set systems

Abstract

Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an O((tlogt)1/2) discrepancy bound in the regime when n \xe2\x89\xa4 m and an O(1) bound when n\xe2\x89\xabmt. In this paper, we give a tight O(\xe2\x88\x9at) bound for the entire range of n and m, under a mild assumption that t=\xce\xa9(log log m)2. The result is based on two steps. First, applying the partial coloring method to the case when n=mlogO(1)m and using the properties of the random set system we show that the overall discrepancy incurred is at most O(\xe2\x88\x9at). Second, we reduce the general case to that of n\xe2\x89\xa4mlogO(1)m using LP duality and a careful counting argument