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≤m and an O(1) bound
when n≫mt.
In this paper, we give a tight O(t) bound for the entire range of
n and m, under a mild assumption that t=Ω(loglogm)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(t).
Second, we reduce the general case to that of n≤mlogO(1)m using LP
duality and a careful counting argument