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Claim. For Problem 747, the threshold is of order $n\log n$, $\ell(n)\asymp n\log n$: there are constants $0<c<C$ such that a random $3$-uniform hypergraph on $3n$ vertices with $Cn\log n$ edges contains $n$ vertex-disjoint edges with probability tending to $1$, while with $cn\log n$ edges it almost surely has an isolated vertex and so contains no such matching. The upper bound is Corollary 2.6 of Factors in random graphs (card): the threshold for a perfect matching in the random $k$-uniform hypergraph on $N$ vertices with edge probability $p$ is $p\asymp N^{-k+1}\log N$, which for $k=3$ and $N=3n$ is $\Theta(n\log n)$ edges; the lower bound is the elementary requirement that every vertex lie in an edge. The corollary is the single-edge case of the paper's Theorem 2.5, the hypergraph form of its main theorem on $H$-factors in $G(N,p)$ for strictly balanced $H$, proved by the entropy and counting method the authors introduce. The library card records the statements; the corpus holds no review of the proof. The sharp constant was later supplied by Kahn's asymptotic for Shamir's problem.

Covers. The order of magnitude of the threshold, $\ell(n)=\Theta(n\log n)$; the constant is left open and is fixed by Kahn's asymptotic for Shamir's problem.

Acceptance. Refereed: Johansson, A., Kahn, J. and Vu, V., Factors in random graphs, Random Structures Algorithms 33 (2008), no. 1, 1–28, published online 2008-05-20; the preprint is arXiv:0803.3406, posted 2008-03-24, the date of this page. Reviewed: the site's curator, Thomas Bloom, labels the problem solved and credits [JKV08] with the threshold $\ell(n)\asymp n\log n$, independently of its authors. No formalization is recorded.