Discrepancy and in -approximations for bounded VC-dimension

Let (X, R) be a set system on an n-point set X. For a two-coloring on X, its discrepancy is defined as the maximum number by which the occurrences of the two colors differ in any set in R. It is shown that if for any m-point subset Y contained in X the number of distinct subsets induced by R on Y is bounded by O(m/sup d/) for a fixed integer d is a coloring with discrepancy bounded by O(n/sup 1/2-1/2d/ (log n)/sup 1+1/2d/). Also, if any subcollection of m sets of R partitions the points into at most O(m/sup d/) classes, then there is a coloring with discrepancy at most O(n/sup 1/2-1/2d/ n). These bounds imply improved upper bounds on the size of in -approximations for (X, R). All of the bounds are tight up to polylogarithmic factors in the worst case. The results allow the generalization of several results of J. Beck (1984) bounding the discrepancy in certain geometric settings to the case when the discrepancy is taken relative to an arbitrary measure.