Dirac-type theorems in random hypergraphs

TL;DR

This paper establishes a transference theorem for Dirac-type perfect matching results in random hypergraphs, linking minimum degree conditions to the existence of perfect matchings in a probabilistic setting.

Contribution

It introduces a transference theorem for Dirac-type theorems in random hypergraphs, using a non-constructive absorbing method to relate minimum degree thresholds to perfect matchings.

Findings

Random hypergraphs typically have perfect matchings under certain minimum degree conditions.

The proof employs a non-constructive absorbing method.

The results connect classical Dirac thresholds to probabilistic hypergraph models.

Abstract

For positive integers $d<k$ and $n$ divisible by $k$, let $m_{d}(k,n)$ 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 $m_{1}(2,n)=\lceil n/2\rceil$. However, in general, our understanding of the values of $m_{d}(k,n)$ 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 $\varepsilon>0$ 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 degree at least $(1+\varepsilon)m_{d}(k,n)p$ 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 $m_{d}(k,n)$ without actually knowing its value.