Asymptotics for Shamir's Problem

TL;DR

This paper determines the asymptotic threshold for the appearance of perfect matchings in random r-uniform hypergraphs and initiates a proof of the hitting time conjecture, extending previous results from 2008.

Contribution

It establishes the precise asymptotic threshold for perfect matchings in random hypergraphs and begins proving the hitting time conjecture, advancing understanding of random hypergraph processes.

Findings

Threshold for perfect matchings is (1+ε)(n/r)log n with high probability.

Asymptotic probability approaches 1 for M above the threshold.

Hitting time for the emergence of a perfect matching is asymptotically 1.

Abstract

For fixed $r\geq 3$ and $n$ divisible by $r$, let ${\mathcal H}={\mathcal H}^r_{n,M}$ be the random $M$-edge $r$-graph on $V=\{1,\ldots ,n\}$; that is, ${\mathcal H}$ is chosen uniformly from the $M$-subsets of ${\mathcal K}:={V \choose r}$ ($:= \{\mbox{$r$-subsets of $V$}\}$). Shamir's Problem (circa 1980) asks, roughly, for what $M=M(n)$ is ${\mathcal H}$ likely to contain a perfect matching (that is, $n/r$ disjoint $r$-sets)? In 2008 Johansson, Vu and the author showed that this is true for $M>C_rn\log n$. The present paper has two purposes. First, it establishes the asymptotically correct version of the 2008 result: Theorem 1. For fixed $\epsilon>0$ and $M> (1+\epsilon)(n/r)\log n$, $P({\mathcal H} ~\mbox{contains a perfect matching})\rightarrow 1 $ as $n\rightarrow\infty$. Second, it begins a proof of the definitive ``hitting time" statement: Theorem 2. If $A_1, \ldots ~$ is a uniform permutation of ${\mathcal K}$, ${\mathcal H}_t=\{A_1,\ldots ,A_t\}$, and $T=\min\{t:A_1\cup \cdots\cup A_t=V\},$ then $P({\mathcal H}_T ~\mbox{contains a perfect matching})\rightarrow 1 $ as $n\rightarrow\infty$. It is shown here that Theorem 2 follows from a conditional version of Theorem 1 that will be proved elsewhere. The key ideas in that proof are similar to those for Theorem 1, but the argument is a longer story, and it has seemed best to give the present separate proof of Theorem 1, in which those ideas may appear more clearly.