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Claim. Theorem 1.2 of the paper: let $s>k\ge5$ with $s>101k^3$, and let $(s+1)k\le n<(s+1)(k+1/(100k))$; then every family of $k$-subsets of an $n$-set with matching number at most $s$ has at most $\binom{(s+1)k-1}{k}$ members, the size of the clique of all $k$-sets inside an $((s+1)k-1)$-set. In the notation of Problem 1020, with $r$ for the uniformity and $k-1$ for the matching number,

$$f(n;r,k)=\binom{rk-1}{r} \qquad\Bigl(r\ge5,\ k-1>101r^3,\ rk\le n<k\bigl(r+\tfrac{1}{100r}\bigr)\Bigr),$$

the conjectured value in that range, where the clique term is the larger. The proof follows Frankl's framework of shifted families and traces on the first $rk-1$ elements. The authors ask whether $1/(100r)$ can be replaced by a small absolute constant. The paper is D. Kolupaev and A. Kupavskii, Erdős matching conjecture for almost perfect matchings, Discrete Math. 346 (2023), Paper No. 113304, carded at Erdős matching conjecture for almost perfect matchings.

Covers. The range $r\ge5$, $k-1>101r^3$ and $rk\le n<k(r+1/(100r))$. The site records the hypothesis as $k>101r^3$. The window replaces the exponentially narrow one of Frankl 2017 at the cost of the lower bound on $k$.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in Discrete Mathematics 346 (2023), no. 4, Paper No. 113304, after its first posting as arXiv:2206.01526 on 2022-06-03. The site labels the problem FALSIFIABLE, an open label, so its commentary, which credits the range to the paper as [KoKu23], is not acceptance and no reviewed is listed. Nothing here rests on this project's own review.