Close the gap in bounds for EF and PROP in group fair division

Determine tight asymptotic bounds, as functions of the number of groups k and the group sizes (n_1, ..., n_k), for the minimum c guaranteeing the existence of (i) an envy-freeness up to c goods (EFc) allocation and (ii) a proportionality up to c goods (PROPc) allocation for additive, monotone utilities over indivisible goods. In particular, ascertain the tight order of the minimum c for EFc in the equal-group-size case n_1 = ... = n_k = n/k by closing the gap between the current lower bound Omega(sqrt(n/k)) and the current upper bound O(sqrt(n)).

Background

The paper proves asymptotically tight lower bounds for multicolor discrepancy and derives improved lower bounds for group fair division guarantees. For consensus 1/k-division, the results are tight; however, for envy-freeness up to c goods (EFc) and proportionality up to c goods (PROPc), the authors only establish improved lower bounds and a refined upper bound for PROPc, leaving a gap in general.

Specifically, for PROPc they obtain an upper bound of O(sqrt(n_1)), where n_1 is the largest group size, and a lower bound of order max_{i* in [k]} (i*/k) sqrt(n_{i*}). For EFc they prove a lower bound of Omega(sqrt(n_1)) while the best known general upper bound remains O(sqrt(n)). The authors highlight the particularly natural case of equal group sizes n_1 = ... = n_k = n/k, where for EFc the current lower bound is Omega(sqrt(n/k)) but the known upper bound is O(sqrt(n)), leaving a factor of sqrt(k) gap. They also remark that when all groups have size 1, EFc = 1 is known, suggesting that sharper upper-bound techniques may be needed.