Exact asymptotic Dirac thresholds for hypergraph perfect matchings

Determine the asymptotic minimum d-degree threshold for perfect matchings in n-vertex k-uniform hypergraphs, and prove or disprove that for all positive integers k and d with d≤k−1 it equals max{1/2, 1−(1−1/k)^(k−d)}.

Background

The paper defines α_d(k) as the limit of the normalized minimum d-degree threshold guaranteeing a perfect matching in k-uniform hypergraphs. It explains that existing extremal constructions suggest that the threshold should be determined by two families of hypergraphs, leading to the displayed formula. The conjecture is discussed as a long-standing threshold problem, although the paper notes announced proofs of Feige’s conjecture that would imply this formula conditionally or through subsequent results.