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Statement

Theorem 1.1. For some absolute constant $c>0$, every tree $T$ on $n$ vertices with maximum degree $\Delta(T)\le cn$, whose bipartition classes have sizes $t_1\ge t_2$, has

$$R(T)=\max\{2t_1,\,t_1+2t_2\}-1.$$

Context from the same page: the lower bound $R(T)\ge\max\{t_1+2t_2,2t_1\}-1$ is Burr's (two constructions, Figure 1: disjoint blue cliques on $t_1+t_2-1$ and $t_2-1$ vertices, or on two sets of $t_1-1$ vertices, with all edges between them red); the constant $c$ "is very small due to the use of regularity methods, and is likely very far from optimal"; the double-star examples of Norin, Sun and Zhao show that $c$ "cannot be improved beyond $7/11+o(1)$"; the existence of $c$ answers a question of Stein (2020). For $t_1=2k$, $t_2=k$ the theorem gives $R(T)=4k-1$ for every tree with classes $2k$ and $k$ whose maximum degree is at most $3ck$.

Source. R. Montgomery, M. Pavez-Signé and J. Yan, Ramsey numbers of trees, arXiv:2509.07934v1 (9 September 2025; the PDF is dated September 10, 2025), 59 pages; Theorem 1.1 on p. 2, read on the page image of the retained PDF. Preprint: no journal record was found on 2026-09-17.

Read depth. Claims checked: the statement and the surrounding remarks on p. 2 were read clause by clause on the page image. The proof (Sections 2--7, pp. 3--59) was not read.

Proof pointer

Section 2.1 (p. 3) divides the proof into a stability part (Sections 4--5, regularity embedding lemmas and four stages of embedding attempts, ending either with a monochromatic $T$ or with a coloring close to one of Burr's two constructions) and an extremal part (Sections 6--7, one section per construction), which the authors call "rather involved"; Section 3 outlines the stability part.

Dependencies

External inputs named in the introduction: Haxell, Łuczak and Tingley's 2002 structure in the reduced graph, Szemerédi's regularity lemma, and the example of Komlós, Sárközy and Szemerédi that makes the extremal part delicate.

Bears on

• Problem 549: the positive restricted case. For trees with classes $k$ and $2k$ the equality $R(T)=4k-1$ holds whenever $\Delta(T)\le3ck$, and the double stars show the degree condition cannot be removed.
• Problem 547: the exact value for trees with $\Delta(T)\le cn$, which lies below the problem's bound since $\max\{2t_1,t_1+2t_2\}-1\le2n-2$ for $t_1+t_2=n$, $t_2\ge1$. The paragraph after the theorem (p. 2, read on the page image) records that Burr and Erdős "conjectured in 1976 that for any $n$-vertex tree $T$, $R(T)\le2n-2$ when $n$ is even and $R(T)\le2n-3$ when $n$ is odd, or in other words $R(T)\le R(K_{1,n-1})$", that Zhao showed this for all large even $n$ in 2011, and that it "follows directly from the Erdős-Sós conjecture", hence for large $n$ from the announced proof of that conjecture for large trees; these are the paper's attributions, not results proved in it.