Sharp exponential bound for the hypergraph parameter α(q)

Prove or disprove that the hypergraph parameter α(q), defined through anti-chains of free pairs in (q,p) hypergraphs, satisfies α(q) ≤ (2 + o(1))^q.

Background

The paper establishes the upper bound α(q) ≤ (2.0922 + o(1))q for general hypergraphs and the sharper bound α(q) ≤ (2 + o(1))q when the hypergraph is restricted to graphs. The authors report only minor improvements for general hypergraphs and leave open whether the graph-like exponential base can be attained in the general case.

Resolving this question would sharpen the bound on n_c and could improve the constant in the asymptotic lower bound for μ(n) for monotone valuations over goods.

References

We were only able to obtain minor improvements over the bounds of Lemma~\ref{lem:alpha}, and leave it as an open question whether $\alpha(q) \le (2 + o(1))q$.

— On MMS allocations with few items  (2610.06125 - Feige, 5 Oct 2026) in Section Discussion; Definition alpha and Lemma alpha