The hitting time of clique factors

TL;DR

This paper extends a recent hitting time result from hypergraphs to clique factors in random graphs, showing that the emergence of a clique factor coincides with the last vertex joining a clique, using novel coupling techniques.

Contribution

It transfers a hypergraph hitting time result to clique factors in random graphs and hypergraphs, introducing new coupling methods.

Findings

Clique factors appear exactly when the last vertex joins a clique of size r.

The result holds for both random graphs and s-uniform hypergraphs.

New coupling techniques extend previous methods.

Abstract

In a recent paper, Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let $r \ge 3 $ and let $n$ be divisible by $r$. Then, in the random $r$-uniform hypergraph process on $n$ vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we transfer this hitting time result to the setting of clique factors in the random graph process: At the time that the last vertex joins a copy of the complete graph $K_r$, the random graph process contains a $K_r$-factor. Our proof draws on a novel sequence of couplings, extending techniques of Riordan and the first author. An analogous result is proved for clique factors in the $s$-uniform hypergraph process ($s \ge 3$).