Discrepancy in different numbers of colors

In this article, we investigate the interrelation between the discrepancies of a given hypergraph in different numbers of colors. Being an extreme example we determine the multi-color discrepancies of the k-balanced hypergraph Hnk on partition classes of (equal) size n. Let c, k, n ∈ N. Set k0:=k mod c and bnkc := (n - [n/[c/k]])k/c. For the discrepancy in c colors we show bnk0c ≤ disc(Hnk, c) < bnk0c + 1, if k0 ≠ 0, and disc(Hnk, c)= 0, if c divides k. This shows that, in general, there is little correlation between the discrepancies of Hnk in different numbers of colors. If c divides k though, disc(H,c) ≤ (k/c)disc(H,k) holds for any hypergraph H.