Abstract One of the foundational theorems of extremal graph theory is Dirac’s theorem, which says that if an n -vertex graph G has minimum degree at least n /2, then G has a Hamilton cycle, and therefore a perfect matching (if n is even). Later work by Sárközy, Selkow and Szemerédi showed that in fact Dirac graphs have many Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph G (in terms of an entropy-like parameter of G). In this paper we extend Cuckler and Kahn’s result to perfect matchings in hypergraphs. For positive integers d d k, and for n divisible by k, let m₃ (k, n) m d (k, n) be the minimum d -degree that ensures the existence of a perfect matching in an n -vertex k -uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of m₃ (k, n) m d (k, n), but we are nonetheless able to prove an analogue of the Cuckler–Kahn theorem, showing that if an n -vertex k -uniform hypergraph G has minimum d -degree at least (1+) m₃ (k, n) (1 + γ) m d (k, n) (for any constant >0 γ > 0), then the number of perfect matchings in G is controlled by an entropy-like parameter of G. This strengthens cruder estimates arising from work of Kang–Kelly–Kühn–Osthus–Pfenninger and Pham–Sah–Sawhney–Simkin.
Kwan et al. (Wed,) studied this question.