Random triangles in random graphs

TL;DR

This paper completes the understanding of the threshold for the appearance of triangle and larger clique factors in random graphs, building on recent hypergraph and probabilistic combinatorics results.

Contribution

It extends Riordan's hypergraph embedding results to the case r=3, establishing a sharp threshold for triangle factors in G(n,p).

Findings

Sharp threshold for triangle factors in G(n,p) established.

Extension of hypergraph embedding techniques to r=3 case.

Implication of Kahn's hypergraph matching bounds for triangle factors.

Abstract

In a recent paper, Oliver Riordan shows that for $r \ge 4$ and $p$ up to and slightly larger than the threshold for a $K_r$-factor, the hypergraph formed by the copies of $K_r$ in $G(n,p)$ contains a copy of the binomial random hypergraph $H=H_r(n,\pi)$ with $\pi \sim p^{r \choose 2}$. For $r=3$, he gives a slightly weaker result where the density in the random hypergraph is reduced by a constant factor. Recently, Jeff Kahn announced an asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem for all $r \ge 3$. With Riordan's result, this immediately implies an asymptotically sharp bound for the threshold of a $K_r$-factor in $G(n,p)$ for $r \ge 4$. In this note, we resolve the missing case $r=3$ by modifying Riordan's argument. This means that Kahn's result also implies a sharp bound for triangle factors in $G(n,p)$.