On the discrepancy of coloring finite sets

Abstract

Given a subset S of {1,…,n} and a map X:{1,…,n}→{−1,1}, (i.e. a coloring of {1,…,n} with two colors, say red and blue) define the discrepancy of S with respect to X to be dX(S)=|∑i∈SX(i)| (the difference between the reds and blues on S). Given n subsets of {1,…,n}, a question of Erdos was to find a coloring of {1,…,n} which simultaneously minimized the discrepancy of the n subsets. We give new and simple proofs of some of the results obtained previously on this problem via an inequality for vectors.