Discrepancy of symmetric products of hypergraphs

For a hypergraph ℋ =(V,ℰ), its d-fold symmetric product is defined to be Δ^dℋ = (V^d, {E^d\E ∈ ℰ). We give several upper and lower bounds for the c-color discrepancy of such products. In particular, we show that the bound disc(Δ^dℋ, 2) ≤ disc(ℋ, 2) proven for all d in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of Cartesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp. cannot be extended to more than c = 2 colors. In fact, for any c and d such that c does not divide d\, there are hypergraphs having arbitrary large discrepancy and disc(Δ^dℋ,c) = Ω _d(disc(ℋ, c)^d). Apart from constant factors (depending on c and d), in these cases the symmetric product behaves no better than the general direct product ℋ^d, which satisfies disc(ℋ^d,c) = O_c,d(disc(ℋ,c)^d).