Discrepancy of Products of Hypergraphs

For a hypergraph $\mathcal{H} = (V,\mathcal{E})$, its $d$―fold symmetric product is $\Delta^d \mathcal{H} = (V^d,\{ E^d | E \in \mathcal{E} \})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound $\textrm{disc}(\Delta^d \mathcal{H},2) \leq \textrm{disc}(\mathcal{H},2)$ 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 $\textrm{disc}(\Delta^d \mathcal{H},c) = \Omega_d(\textrm{disc}(\mathcal{H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product $\mathcal{H}^d$, which satisfies $\textrm{disc}(\mathcal{H}^d,c) = O_{c,d}(\textrm{disc}(\mathcal{H},c)^d)$

Discrepancy of products of hypergraphs

For a hypergraph H = (V, E), its d–fold symmetric product is ∆ d H = (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( ∆ d H, 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( ∆ d H, c) = Ωd(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) = Oc,d(disc(H, c) d)