---

Claim. For all $n\ge4$ and $k\ge4n+2$,

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

in the letters of Problem 551. This is Theorem 1 of V. Nikiforov's preprint arXiv:math/0404501v1, first posted on 27 April 2004, the date this page is named by, whose journal version is The cycle-complete graph Ramsey numbers, Combin. Probab. Comput. 14 (2005), no. 3, 349--370, cited as [Ni05] on the problem page; the paper writes $r(C_p,K_r)$ with $p$ the cycle length and states the theorem for $r\ge4$ and $p\ge4r+2$ (p. 2 of the preprint, whose statement numbers the journal version does not share). Its introduction records the earlier ranges $k\ge n^2-2$ of Bondy and Erdős and $k\ge n^2-2n$ of Schiermeyer and the cases $n=4,5,6$ as settled. The proof finds, in a $C_k$-free graph of order $(k-1)(n-1)+1$ with independence number below $n$, a Hamiltonian cycle with many chords whose paths have many consecutive lengths and assembles a cycle of the forbidden length; it is recorded on the result page Theorem 1 of the library home nikiforov_2005_cycle_complete_graph_ramsey_numbers. The concluding remarks (p. 22) say that the method reaches $k\ge3n+9$ except for one lemma and that it seems more refinement could reach $k\ge2n+o(n)$, and conjecture a polynomial threshold.

Covers. The pairs $(k,n)$ with $n\ge4$ and $k\ge4n+2$, infinitely many for each $n$; this contains the range of Bondy and Erdős 1973 for every $n\ge5$. Not covered: $n=3$, which is classical, and the pairs with $n\le k\le4n+1$, of which Keevash, Long and Skokan 2021 leaves finitely many $n$; the finite residue is stated on that page.

Depends on. No page of this wiki: the theorem rests on the paper's own lemmas, the Erdős--Gallai theorem on long paths, the earlier results for clique orders 4, 5 and 6 (Yang, Huang and Zhang; Bollobás et al.; Schiermeyer) that start its induction (preprint p. 5), and, in the proof of Lemma 5 (p. 12), a theorem of Dirac and Corollary 2.13 of Bondy's handbook chapter.

Acceptance. Refereed: the paper is a journal publication in Combinatorics, Probability and Computing, volume 14, issue 3 (2005), published 11 April 2005, the refereed evidence. 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 arXiv preprint; the journal text, which is paywalled, is not compared, the proof is not checked, and nothing is independently reviewed by this corpus.