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Claim. There is $n_0$ such that every tree $T$ on $n\ge n_0$ vertices has

$$R(T)\le2n-2.$$

This is Corollary 2.2 (p. 5) of the library's [[../library/extremal_graph_theory/zhao_2011_proof_n_2_n_2_n_2_conjecture_large_n/_index|source card]], deduced from the paper's main result, [[../library/extremal_graph_theory/zhao_2011_proof_n_2_n_2_n_2_conjecture_large_n/theorem_1_6|Theorem 1.6]]: for $n\ge n_0$, a graph on $n$ vertices in which at least $\lceil n/2\rceil$ vertices have degree at least $\lceil n/2\rceil$ contains every tree with at most $\lfloor n/2\rfloor$ edges. Applied to the color class of a two-colored $K_{2n-2}$ in which at least $n-1$ vertices have degree at least $n-1$ (one of the two classes does, since every vertex has total degree $2n-3$; p. 5, the paragraph before the corollary), this yields a monochromatic copy of every tree on $n$ vertices. The paper does not claim the odd-$n$ refinement $2n-3$. A Lean development in Boris Alexeev's repository, linked above, formalizes this corollary as eventually_erdos_547, following the paper's proof, with modules named after its Claims 6.10 and 6.15; this corpus has not built it, so it gives no formalized evidence.

Covers. The corrected Statement of Problem 547 for every tree whose order $n$ is at least the paper's threshold $n_0$, which the paper does not make explicit. What remains is the bound for the finitely many orders below $n_0$; the site's label DECIDABLE describes this reduction, and its commentary cites the paper for the large-$n$ bound. The bound for every $n\ge2$ is the subject of the accepted claim page [[problems/ramsey_theory/E0547/claims/2026_09_03_adamczewski|the 2026 claim]].

Depends on. Nothing in this wiki; the result is the paper's own corollary.

Acceptance. Not reviewed: the site's curator, T. F. Bloom, credits the paper in the problem's commentary with the bound for all large $n$, but Bloom's label DECIDABLE settles neither the problem nor any part of it, so the credit is not acceptance evidence. Refereed: Electronic Journal of Combinatorics 18 (2011), no. 1, Paper 27, 61 pp.; submitted 6 June 2008, accepted 22 January 2011 and published 4 February 2011 (p. 1; the journal's foot is printed on every page), the date this page is named by.

Read depth. Theorem 1.6 and Corollary 2.2 were read clause by clause in the text layer of pp. 2 and 5, and Theorem 1.6 again on the page image; the proof (Sections 3--7 and the appendix, pp. 5--61, by the Regularity Lemma and tree embedding lemmas) was not read, and nothing is independently reviewed in this corpus.