Factors in random graphs

TL;DR

This paper determines the threshold probability for the existence of an $H$-factor in Erdős-Rényi random graphs, extending to hypergraphs and solving Shamir's problem for perfect matchings.

Contribution

It provides the exact threshold for $H$-factors in random graphs for all strictly balanced $H$, and extends the method to hypergraphs, solving a longstanding problem.

Findings

Determined the threshold for $H$-factors in Erdős-Rényi graphs.

Extended the method to hypergraphs.

Solved Shamir's problem for perfect matchings in hypergraphs.

Abstract

Let $H$ be a fixed graph on $v$ vertices. For an $n$-vertex graph $G$ with $n$ divisible by $v$, an $H$-{\em factor} of $G$ is a collection of $n/v$ copies of $H$ whose vertex sets partition $V(G)$. In this paper we consider the threshold $th_{H} (n)$ of the property that an Erd\H{o}s-R\'enyi random graph (on $n$ points) contains an $H$-factor. Our results determine $th_{H} (n)$ for all strictly balanced $H$. The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random $k$-uniform hypergraph, solving the well-known "Shamir's problem."