Tight Lower Bound for Multicolor Discrepancy

We prove the following asymptotically tight lower bound for $k$-color discrepancy: For any $k\ge 2$, there exists a hypergraph with $n$ hyperedges such that its $k$-color discrepancy is at least $\mathrm{\Omega }(\sqrt{n})$. This improves on the previously known lower bound of $\mathrm{\Omega }(\sqrt{n\mathrm{/}\log k})$ due to Caragiannis et al. (arXiv:2502.10516). As an application, we show that our result implies improved lower bounds for group fair division.