Multi–color Discrepancies

In this article we introduce combinatorial multi-color discrepancies and generalize several classical results from 2–color discrepancy theory to c colors (c ≥ 2). We give a recursive method that constructs c–colorings from approximations of 2–color discrepancies. This method works for a large class of theorems like the ‘six standard deviations’ theorem of Spencer, the Beck–Fiala theorem, the results of Matoušek, Welzl and Wernisch and Matoušek for bounded VC–dimension and Matoušek’s and Spencer’s upper bound for the arithmetic progressions. In particular, the c–color discrepancy of an arbitrary hypergraph (n vertices, m hyperedges) is O( √ n c logm). If m = O(n), then this bound improves to O( √ n c log c). On the other hand there are examples showing that discrepancy in c colors can not be bounded in terms of two-color discrepancies in general, 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 floating colors via tensor products of matrices to multicolorings and prove multi-color versions of the the Beck–Fiala theorem and the Barany– Grunberg theorem. Using properties of the tensor product we derive a lower bound for the c–color discrepancy of general hypergraphs. For the hypergraph of arithmetic progressions in {1, . . . , n} this yields a lower bound of 1 25 √ c 4 √ n for the discrepancy in c colors. The recursive method shows an upper bound of O(c−0.16 4 √ n).