---

Claim. The preprint A Near-Optimal Linear Range for the Erdős Matching Conjecture by Mengyu Cao, Hong Liu and Haixiang Zhang, arXiv:2608.19118 (posted 2026-08-19, revised 2026-09-07), states that for every fixed $k\ge2$ there is $s_0(k)$ such that, whenever $s\ge s_0(k)$ and $n\ge(k+1)s$, every family of $k$-subsets of an $n$-set with matching number at most $s$ has at most $\binom nk-\binom{n-s}{k}$ members, with equality only for the family of $k$-sets meeting a fixed $s$-set; for $k\ge5$ the abstract gives the coefficient $k+0.6$ in place of $k+1$. In the notation of Problem 1020, with $r$ for the uniformity and $k-1$ for the matching number, for every fixed $r\ge3$ there is $s_0(r)$ such that

$$f(n;r,k)=\binom nr-\binom{n-k+1}{r} \qquad(k-1\ge s_0(r),\ n\ge(r+1)(k-1)),$$

the conjectured value in that range. Since the covering term is the larger exactly from about $n=(r+1)k$ on, this is close to the whole range of the conjecture's second term for large $k$. The abstract describes a stability theorem and probabilistic rigidity arguments. A reader posted the preprint on the site's discussion thread on 2026-08-20.

Covers. Every fixed $r\ge3$ for $k-1\ge s_0(r)$ and $n\ge(r+1)(k-1)$, lowering the coefficient $\tfrac53$ of Frankl and Kupavskii 2022 to $1+1/r$. It says nothing about small $k$ or about the clique range.

Depends on. No page of this wiki.

Standing. Claimed. The preprint is not refereed, no proof claim was registered on the site's proof-claims tab, and the site's label and commentary, last edited on 28 December 2025, do not mention it. The proof is not verified by this corpus.