Close the sqrt(ln k) gap for multi-color discrepancy and consensus 1/k-division

Determine tight bounds for the minimum multi-color discrepancy DISC(n,k) and for the minimum d ensuring a consensus 1/k-division CD(n,k) by closing the remaining Θ(√ln k) factor gap between the best known upper bounds O(√n) and the new lower bounds Ω(√(n/ln k)) for set systems with n sets and k colors and for fair division instances with n agents and k bundles, respectively.

Background

The paper proves a new lower bound for multi-color discrepancy: there exists a set system with n sets such that any k-coloring has discrepancy at least Ω(√(n/ln k)). Via a known reduction, this immediately yields the same lower bound for consensus 1/k-division up to d items (CDd).

Existing upper bounds for these quantities are O(√n), leaving a multiplicative gap of Θ(√ln k). The authors explicitly identify closing this gap as an open problem.