We investigate the problem to color the vertex set of a hypergraph H = (X, £) with a fixed number of colors in a balanced manner, i.e., in such a way that all hyperedges contain roughly the same number of vertices in each color (discrepancy problem). We show the following result: Suppose that we are able to compute for each induced subhypergraph a coloring in c 1  colors having discrepancy at most D. Then there are colorings in arbitrary numbers c2 of colors having discrepancy at most 11 c 2  1 D. A c 2 -coloring having discrepancy at most 11/10 c 2  1 D + 3c -k  1 |X| can be computed from (c 1  - 1)(c 2  - 1)k colorings in c 1  colors having discrepancy at most D with respect to a suitable subhypergraph of H. A central step in the proof is to show that a fairly general rounding problem (linear discrepancy problem in c2 colors) can be solved by computing low-discrepancy c 1 -colorings.
['Benjamin Doerr']
Balanced coloring: Equally easy for all numbers of colors?