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Claim. For $k,\ell\ge2$ let $Q_{k,\ell}$ be the tree obtained from a path $P_4$ on four vertices by appending a star $K_{1,k-2}$ at one end and a star $K_{1,\ell-2}$ at the other; its classes have $k$ and $\ell$ vertices. Lemma 4.1 (p. 252) of S. A. Burr and P. Erdős, Extremal Ramsey theory for graphs, Utilitas Math. 9 (1976), 247--258, which writes the tree as $S_{k,\ell}$, states that

$$r(Q_{k,\ell})=\max(2k-1,\,k+2\ell-1)\qquad(k\ge\ell\ge2).$$

At classes $2k$ and $k$ this is $r(Q_{2k,k})=4k-1$ for every $k\ge2$: the equality of Problem 549 holds for the tree made of a path on four vertices with stars on $2k-1$ and $k-1$ vertices at its ends. The lower bound is Burr's 1974 coloring bound; the upper bound takes a red $K_{1,k-1}\cup K_{1,\ell}$, which exists by a result of Rosta, $r(K_{1,k-1}\cup K_{1,\ell})=\max(2k-1,k+2\ell-1)$, quoted in the paper from Burr's 1974 survey, and completes it to a monochromatic $Q_{k,\ell}$ by a short case analysis. The statement is recorded on the result page Lemma 4.1 of the library home burr_1976_extremal_ramsey_theory_graphs; Erdős, Faudree, Rousseau and Schelp (1982, p. 284) credit the result to this paper.

Covers. For every $k\ge2$, the tree with classes $2k$ and $k$ formed from $P_4$ by the stars $K_{1,2k-2}$ and $K_{1,k-2}$ at its ends has $R(T)=4k-1$. Every other tree with these classes is outside it; the problem's equality fails for the double stars, as the full claim pages record.

Depends on. Nothing in this wiki; the upper bound uses Rosta's result on $r(K_{1,k-1}\cup K_{1,\ell})$, unpublished and quoted through Burr's 1974 survey (Lecture Notes in Math. 406), which this corpus does not hold.

Acceptance. Refereed: the paper is a journal publication in Utilitas Mathematica, volume 9 (1976), received 5 November 1974, the refereed evidence; the record carries no month, so this page is dated to the first day of the year. The site's curator lists this family in the commentary, but the label DISPROVED credits the disproof, not this case, so reviewed is not listed.