The hitting time of clique factors
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 ($s \ge 3$).