The hitting time of clique factors

In \cite{kahn2022hitting}, Kahn gave the strongest possible, affirmative, answer to Shamir‘s problem, which had been open since the late 1970s: Let r \geq 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 prove the analogue of this result for 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 which embeds the random hypergraph process into the cliques of the random graph process. An analogous result is proved for clique factors in the s-uniform hypergraph process (s \geq 3).