Random cliques in random graphs and sharp thresholds for $F$-factors

TL;DR

This paper investigates the distribution of cliques in random graphs near the threshold for perfect clique packings, establishing sharp probabilistic bounds and extending results to hypergraphs and other structures.

Contribution

It proves that for each r ≥ 4, the copies of K_r in G(n,p) are randomly distributed near the threshold, extending sharp threshold results to hypergraph factors and other graphs.

Findings

Distribution of K_r copies is random near the threshold

Sharp bounds for K_r-factor thresholds are established

Results extend to hypergraphs and other structures

Abstract

We show that for each $r\ge 4$, in a density range extending up to, and slightly beyond, the threshold for a $K_r$-factor, the copies of $K_r$ in the random graph $G(n,p)$ are randomly distributed, in the (one-sided) sense that the hypergraph that they form contains a copy of a binomial random hypergraph with almost exactly the right density. Thus Jeff Kahn's recent asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem implies a corresponding bound for the threshold for $G(n,p)$ to contain a $K_r$-factor. The case $r=3$ is more difficult, and has been settled by Annika Heckel. We also prove a corresponding result for $K_r^{(t)}$-factors in random $t$-uniform hypergraphs, as well as (in some cases weaker) generalizations replacing $K_r$ by certain other (hyper)graphs.