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Claim. Theorem 1.1 (p. 2) of R. Montgomery, M. Pavez-Signé and J. Yan, Ramsey numbers of trees, arXiv:2509.07934v1, states: "There exists a constant $c>0$ such that the following holds. Any $n$-vertex tree $T$ with $\Delta(T)\leq cn$ and bipartition classes of sizes $t_1\geq t_2$ satisfies $R(T)=\max\{2t_1,t_1+2t_2\}-1$." For classes $2k$ and $k$, so $n=3k$, this is $R(T)=4k-1$ whenever $\Delta(T)\le3ck$: the equality of Problem 549 for every such tree of small linear maximum degree. The authors say that their $c$ is very small, because the proof uses regularity methods, and that the double stars of Norin, Sun and Zhao show $c$ cannot exceed $7/11+o(1)$. The proof is a stability analysis: colorings far from Burr's two extremal constructions are handled with Szemerédi's regularity lemma, building on Haxell, Łuczak and Tingley, and colorings close to them by a separate extremal analysis. The statement is recorded on the result page Theorem 1.1 of the library home montgomery_2025_ramsey_numbers_trees.

Covers. Every tree with classes $2k$ and $k$ and maximum degree at most $3ck$, for the paper's constant $c$, once $k$ is large enough for such a tree to exist. The trees of larger maximum degree are outside it; among them the problem's equality fails for the double stars and holds for the brooms and for Burr and Erdős's trees, as the other claim pages record.

Depends on. Nothing in this wiki; the proof uses Szemerédi's regularity lemma and the reduced-graph structure of Haxell, Łuczak and Tingley (2002).

Standing. Claimed: the paper is an arXiv preprint, posted on 9 September 2025, the only version, with no journal record found on 2026-09-17; its 59-page proof is not checked in this corpus. The site's curator lists the result in the commentary, but the label DISPROVED credits the disproof, not this case, so reviewed is not listed.