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Claim. For every graph $H$ with $q$ edges and no isolated vertices,

$$R(K_3,H)\le2q+1.$$

With $m=q$ this is the case $k=3$ of Problem 570, since $\lfloor(3-1)/2\rfloor=1$, and it holds for every $m$ rather than only for large $m$. The paper states that the bound settles Harary's conjecture and is best possible as a function of $q$. The theorem is paged as the main theorem of the library's source card, which is based on the seven-page author-hosted manuscript. Sidorenko proved the same theorem independently (Sidorenko 1993).

Covers. The case $k=3$, for every $m\ge1$. Nothing about any other cycle length.

Depends on. Nothing in this wiki; the result rests on the cited paper, whose proof takes the case of minimum degree 1 from Sidorenko's 1991 note (J. Graph Theory 15 (1991), 15--17), cited on the manuscript's p. 2.

Acceptance. Reviewed: the site's curator, T. F. Bloom, labels the problem proved and credits the case $k=3$ to this paper and to Sidorenko (its key [GoKl94]; page last edited 16 January 2026, accessed 2026-09-08 for the problem page), and the 2026 preprint of Cambie, Freschi, Morawski, Petrova and Pokrovskiy states the theorem as its Theorem 1 with the same attribution. Refereed: W. Goddard and D. J. Kleitman, An upper bound for the Ramsey numbers $r(K_3,G)$, Discrete Math. 125 (1994), no. 1--3, 177--182, in the February 1994 issue (Crossref), the month this page is dated by; the day is a placeholder.

Read depth. The statement on p. 1 of the author-hosted manuscript is checked against the problem's formula; the proof on pp. 2--6, an induction on $q$ organized by the minimum degree of $H$, is not checked, nor is the manuscript compared with the journal text. Nothing is independently reviewed in this corpus.