In [Trans. Am. Math. Soc. 375 (2022), no. 1, 627–668], Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let $r⩾3$ 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 ($s⩾3$).

中文翻译：

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派系因素的打击时间

在[翻译。是。数学。苏克。 375（2022），没有。 1, 627–668]，卡恩对自 20 世纪 70 年代末以来一直开放的沙米尔问题给出了最强烈、肯定的答案：让$r⩾3$然后让$n$可以被整除$r$。然后，在随机$r$-统一超图过程$n$顶点，一旦最后一个孤立顶点消失，就会出现完美匹配。在目前的工作中，我们证明了随机图过程中派系因子的结果的类似物：在最后一个顶点加入完整图的副本时$K_r$，随机图过程包含$K_r$-因素。我们的证明利用了一种新颖的耦合序列，它将随机超图过程嵌入到随机图过程的派系中。对于派系因素，也证明了类似的结果$s$- 统一超图过程（$s⩾3$）。