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Claim. A family of $k$-subsets of an $n$-set with no $s+1$ pairwise disjoint members has at most $\binom nk-\binom{n-s}{k}$ members provided $n\ge\tfrac53sk-\tfrac23s$ and $s$ is sufficiently large, as the paper's abstract states its main theorem. In the notation of Problem 1020, with $r$ for the uniformity and $k-1$ for the matching number, there is $k_0$ such that

$$f(n;r,k)=\binom nr-\binom{n-k+1}{r} \qquad\Bigl(k\ge k_0,\ n\ge\tfrac53(k-1)r-\tfrac23(k-1)\Bigr),$$

the conjectured value in that range. The paper is P. Frankl and A. Kupavskii, The Erdős Matching Conjecture and concentration inequalities, J. Combin. Theory Ser. B 157 (2022), 366–400.

Covers. The range $n\ge\tfrac53(k-1)r-\tfrac23(k-1)$ for $k\ge k_0$. It lowers the coefficient $2$ of Frankl 2013 to $\tfrac53$ for large $k$; the coefficient $r+1$ of the conjecture's crossover, $n\ge(r+1)(k-1)$ for $k-1\ge s_0(r)$, is the pending claim on Cao, Liu and Zhang 2026.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in the Journal of Combinatorial Theory, Series B, 157 (2022), 366–400, after its first posting as arXiv:1806.08855 on 2018-06-22. 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.