---

Claim. For every graph $H$ with $m\ge1$ edges and no isolated vertices,

$$R(K_3,H)\le2m+1\le3m,$$

so in the notation of Problem 569 $c_1\le3$. The one-edge graph $K_2$ is eligible and $R(C_3,K_2)=3$, so $c_1\ge3$, and hence $c_1=3$. Goddard and Kleitman proved the same theorem independently (Goddard and Kleitman 1994); the same theorem is the case $k=3$ of Problem 570, recorded on its claim page there, whose account of the statement's sources this page follows. A comment of 27 March 2026 in the site's discussion thread credits the case $k=1$ to this paper and to Goddard and Kleitman.

Covers. The case $k=1$, $c_1=3$. Nothing about $k\ge2$.

Depends on. Nothing in this wiki; the result rests on the cited paper, and the one-edge endpoint is elementary.

Acceptance. Refereed: A. F. Sidorenko, The Ramsey number of an $n$-edge graph versus triangle is at most $2n+1$, J. Combin. Theory Ser. B 58 (1993), no. 2, 185--196, in the July 1993 issue, the month this page is dated by; the day is a placeholder. The site labels the problem OPEN, so its pages are not acceptance.

Read depth. The paper is not held. The statement, with its hypotheses ($n$ edges, no isolated vertices), is read in the zbMATH Open review of the paper (Zbl 0794.05090): "We prove the conjecture of Harary that for any graph $G$ with $n$ edges and without isolated vertices, $r(K_3,G)\leq 2n+1$". Theorem 1 of the 2026 preprint of Cambie, Freschi, Morawski, Petrova and Pokrovskiy attributes the same statement to it. Nothing is independently reviewed in this corpus.