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Statement

"These results suggest a general question. If $T_n$ is any tree with parts of size $n/3$ and $2n/3$ is $r(T_n)=\{4n/3-1\}$?" (printed p. 292, the last sentence of Section II).

The paper writes $\{x\}$ for the least integer $\ge x$ and $[x]$ for the integer part: on p. 286 a broom $B_{k,\ell}$ has parts of sizes $\{\ell/2\}$ and $k+[\ell/2]$, which sum to $k+\ell$. With $n=3k$ the parts are $k$ and $2k$ and the value is $4k-1$, the statement of Problem 549. It is posed as a question, after Theorem 2.2 and the remark on p. 288 that the brooms $B_{k,\ell}$ with $2k\le\ell\le2k+2$ attain $r(B_{k,\ell})=\{4(k+\ell)/3-1\}$, "a specific tree whose Ramsey number is as small as possible".

Source. P. Erdős, R. J. Faudree, C. C. Rousseau and R. H. Schelp, Ramsey numbers for brooms, Congr. Numer. 35 (1982), 283--293; printed p. 292 is PDF p. 10 of the scan, read on the page image (the scan's text layer garbles the braces).

Read depth. Claims checked: the sentence and the notation on pp. 286 and 288 were read on the page images. A question has no proof.

Proof pointer

None; the question is answered in the negative by Norin, Sun and Zhao (2016).

Dependencies

None.

Bears on

• Problem 549: the origin of the problem, posed as a question rather than a conjecture.