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Claim. If $k\ge3$, the largest matching in a $k$-uniform hypergraph on $n$ vertices has exactly $s$ edges and $n>2k^2s/\log k$, then the hypergraph has at most $\binom nk-\binom{n-s}{k}$ edges, and only the family of all $k$-sets meeting a fixed $s$-set attains the bound, as Theorem 1 of the paper states; the abstract prints the range as $n>3k^2s/2\log k$. In the notation of Problem 1020, with $r$ for the uniformity and $k-1$ for the matching number,

$$f(n;r,k)=\binom nr-\binom{n-k+1}{r}\qquad\Bigl(n>\frac{2r^2(k-1)}{\log r}\Bigr),$$

the conjectured value in that range. The paper is P. Frankl, T. Łuczak and K. Mieczkowska, On matchings in hypergraphs, Electron. J. Combin. 19 (2012), no. 2, Paper 42, carded at On matchings in hypergraphs.

Covers. The range $n>2r^2(k-1)/\log r$, which the site records as $n>2kr^2/\log r$. It improves Huang, Loh and Sudakov 2012 by the logarithm and was superseded by the linear range of Frankl 2013.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in the Electronic Journal of Combinatorics, volume 19, issue 2, Paper 42, published on 2012-06-13, the page's date. The site labels the problem FALSIFIABLE, an open label, so its commentary, which credits the range to the paper as [FLM12], is not acceptance and no reviewed is listed. Nothing here rests on this project's own review.