Approximation of Multi-color Discrepancy

In this article we introduce (combinatorial) multi{color discrepancy and generalize some classical results from 2{color discrepancy theory to c colors. We give a recursive method that constructs c{colorings from approximations to the 2{color discrepancy. This method works for a large class of theorems like the six{standard{deviation theorem of Spencer, the Beck{Fiala theorem and the results of Matou sek, Welzl and Wernisch for bounded VC{dimension. On the other hand there are examples showing that discrepancy in c colors can not be bounded in terms of two{color discrepancy even if c is a power of 2. For the linear discrepancy version of the Beck{Fiala theorem the recursive approach also fails. Here we extend the method of oating colors to multi{colorings and prove multi{color versions of the the Beck{Fiala theorem and the Barany{Grunberg theorem.