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Claim. For a family of $k$-subsets of an $n$-set with no $s+1$ pairwise disjoint members, $n\ge(2s+1)k-s$ implies that the family has at most $\binom nk-\binom{n-s}{k}$ members, the paper's main theorem in its own notation. In the notation of Problem 1020, with $r$ for the uniformity and $k-1$ for the matching number,

$$f(n;r,k)=\binom nr-\binom{n-k+1}{r}\qquad(n\ge(2k-1)r-(k-1)),$$

the conjectured value in that range, where the covering term is the larger. The paper is P. Frankl, Improved bounds for Erdős' Matching Conjecture, J. Combin. Theory Ser. A 120 (2013), 1068–1072.

Covers. The range $n\ge(2k-1)r-(k-1)$, about $2rk$. It supersedes the ranges of order $r^2k$ on Huang, Loh and Sudakov 2012 and Frankl, Łuczak and Mieczkowska 2012, and was lowered to about $\tfrac53rk$ for large $k$ on Frankl and Kupavskii 2022.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in the Journal of Combinatorial Theory, Series A, 120 (2013), no. 5, 1068–1072; the record dates the issue to July 2013 and gives no day, so the page is dated to the first day of that month. The site's commentary does not cite the paper and labels the problem FALSIFIABLE, an open label, so no reviewed is listed. Nothing here rests on this project's own review.