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Claim. For all $n\ge3$ and $k\ge n^2-2$,

$$R(C_k,K_n)=(k-1)(n-1)+1,$$

in the letters of Problem 551. This is Theorem 4 (printed p. 52) of J. A. Bondy and P. Erdős, Ramsey numbers for cycles in graphs, J. Combinatorial Theory Ser. B 14 (1973), no. 1, 46--54, cited as [BoEr73] on the problem page; the paper writes $R(C_n,K_r)$ with $n$ the cycle length and states the theorem for $n\ge r^2-2$, and its introduction (p. 47) derives the identity for all large cycle lengths from Theorem 3 before proving this explicit range directly. The proof is an induction on the clique order through Turán's theorem, the Erdős--Gallai bound for long cycles and the paper's own lemmas on chords; it is recorded on the result page Theorem 4 of the library home bondy_1973_ramsey_numbers_cycles_graphs. The site's commentary writes the range as $k>n^2-2$; the paper's is $k\ge n^2-2$.

Covers. The pairs $(k,n)$ with $n\ge3$ and $k\ge n^2-2$, infinitely many for each $n$. For $n=3$ this is $k\ge7$, and $R(C_k,K_3)=2k-1$ for every $k>3$ is classical (quoted from Chartrand and Schuster on p. 47 of the paper). Not covered: the pairs with $n\le k<n^2-2$, which Nikiforov 2005 reduces to $k\le4n+1$ and Keevash, Long and Skokan 2021 to finitely many $n$; the finite residue is stated on that page.

Depends on. No page of this wiki: the theorem and its proof are the paper's own.

Acceptance. Refereed: the paper is a journal publication in the Journal of Combinatorial Theory, Series B, volume 14, number 1 (February 1973), received 28 January 1972, the refereed evidence; the issue carries no day, so this page is dated to the first day of that month. The site's curator, Thomas Bloom, credits the range to this paper in the problem page's commentary, but the site's label DECIDABLE settles neither the problem nor a declared part of it, so that credit is not reviewed evidence. The statement is checked against the paper; the proof is read for its structure only, and nothing is independently reviewed by this corpus.