Asymptotically sharp constant for monotone-goods MMS feasibility

Determine whether, for monotone valuations over goods, the maximin-share feasibility threshold satisfies μ(n) = n + log n + o(log n).

Background

The paper proves that for monotone valuations over goods, μ(n) = n + Θ(log n), with the leading constant bounded below by approximately 0.93 and above by 1 + o(1). The unresolved issue is whether the upper-end constant is asymptotically achievable, namely whether the threshold is n + log n up to a lower-order term.

The question is connected to improving the bounds on the hypergraph parameter α(q) introduced in Definition of the paper’s hypergraph analysis.