Dirac-type theorems in random hypergraphs

Abstract

For positive integers d<k and n divisible by k, let md(k,n) be the minimum d-degree ensuring the existence of a perfect matching in a k-uniform hypergraph. In the graph case (where k=2), a classical theorem of Dirac says that m1(2,n)=⌈n/2⌉. However, in general, our understanding of the values of md(k,n) is still very limited, and it is an active topic of research to determine or approximate these values. In this paper we prove a “transference” theorem for Dirac-type results relative to random hypergraphs. Specifically, for any d<k, any ε>0 and any “not too small” p, we prove that a random k-uniform hypergraph G with n vertices and edge probability p typically has the property that every spanning subgraph of G with minimum d-degree at least (1+ε)md(k,n)p has a perfect matching. One interesting aspect of our proof is a “non-constructive” application of the absorbing method, which allows us to prove a bound in terms of md(k,n) without actually knowing its value.