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Claim. The answer to Problem 747 is $\ell(n)\sim n\log n$. Theorem 1 of Asymptotics for Shamir's problem (card) states that for fixed $r\ge3$ and $\varepsilon>0$, the random $r$-uniform hypergraph on $N$ vertices with $M>(1+\varepsilon)(N/r)\log N$ edges contains a perfect matching with probability tending to $1$. With $r=3$ and $N=3n$ this gives $n$ vertex-disjoint edges once $M>(1+\varepsilon)n\log(3n)$, and with fewer than $(1-\varepsilon)n\log(3n)$ edges some vertex is almost surely isolated, so the threshold is $(1+o(1))n\log n$ and isolated vertices are asymptotically the only obstruction. This sharpens the order-of-magnitude result of Johansson, Kahn and Vu's threshold of order n log n by the same entropy and counting approach with the constant pushed to its correct value. The paper's Theorem 2, the hitting-time statement that the random hypergraph process has a perfect matching as soon as its last isolated vertex disappears, is shown there to follow from a conditional strengthening of Theorem 1 that the paper defers to J. Kahn, Hitting times for Shamir's problem, Trans. Amer. Math. Soc. 375 (2022), no. 1, 627–668 (arXiv:2008.01605); it is not needed for the threshold. The library card records the statements; the corpus holds no review of the proof.

Acceptance. Refereed: Kahn, J., Asymptotics for Shamir's problem, Adv. Math. 422 (2023), Paper No. 109019, 39 pp.; the preprint is arXiv:1909.06834, posted 2019-09-15, the date of this page. Reviewed: the site's curator, Thomas Bloom, labels the problem solved and credits [Ka23] with the asymptotic $\ell(n)\sim n\log n$, independently of its author. No formalization is recorded.