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Claim. With $n_k$ as in Problem 1063, $n_k\le k!$ for every $k\ge3$. The source is J.-M. Monier's solution of Problem 6447, proposed by P. Erdős and J. L. Selfridge, Amer. Math. Monthly 92 (1985), no. 6, 435–436, DOI 10.2307/2322464 (Crossref lists the item as "6447" by Erdős, Selfridge and Monier). No public copy of the solution is available, so the statement is recorded as Guy's B31 reports it, citing the 1985 solution beside $n_k\le k!$ for $k\ge3$. The site's commentary also credits Monier with a proof of the Erdős–Selfridge fact that for $n\ge2k$ some $n-i$ with $0\le i<k$ fails to divide $\binom nk$; that attribution is the site's.

Covers. The upper bound $n_k\le k!$ for $k\ge3$, which gives $\log n_k\le(1+o(1))\,k\log k$. Not covered: any lower bound beyond $2k$, the order of $\log n_k$, and the estimate Erdős and Selfridge asked for. Cambie's sharper bound $n_k\le k\,[1,\ldots,k-1]\le e^{(1+o(1))k}$ was posted as a thread comment and adopted in the site's commentary; it is not a publication and has no page.

Acceptance. A refereed journal publication in The American Mathematical Monthly (refereed). The site's commentary credits the result, but the site labels the problem OPEN, so that credit is not reviewed. The record gives the issue month, June 1985, and no day, so this page is named by the first day of that month.

Depends on. No page of this wiki.