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Claim. For every fixed $k\ge2$ and all sufficiently large odd $n$,

$$R_k(C_n)=2^{k-1}(n-1)+1.$$

At $k=3$ this is $R_3(C_n)=4n-3$ for every odd $n$ beyond a threshold, which is the problem's inequality with equality. The lower bound is the classical doubling construction, recorded in the paper as the Erdős--Graham bound $2^{k-1}(n-1)+1\le R_k(C_n)$ for odd $n>3$; the upper bound is the paper's own proof, which reduces the problem by the regularity method to maximizing the $\ell_1$-norm over a compact set and classifies the extremal points, so the threshold on $n$ is not effective. The statement is paged at Theorem 1.2 of the library's source card. The paper records that the case $k=3$ was first resolved by Kohayakawa, Simonovits and Skokan and describes its own stability theorem as a strengthening that generalizes their main result; the paper gives a second proof of the statement that the page Kohayakawa, Simonovits and Skokan 2005 records.

Covers. The inequality $R_3(C_n)\le4n-3$ for all sufficiently large odd $n$, with equality. Not covered: even $n$ (the page Benevides and Skokan 2008), and the odd $n$ below the unnamed threshold.

Acceptance. Refereed: Exact Ramsey numbers of odd cycles via nonlinear optimisation, Adv. Math. 376 (2021), Paper No. 107444 (the Crossref record). The preprint is arXiv:1608.05705v1 of 19 August 2016, the date this page is named by, and the only version the arXiv listing shows; the journal text was not compared with it, so locators are the preprint's. The site's commentary on this problem does not cite the paper, so no curator credit is listed.

Read depth. Claims checked: Conjecture 1.1, display (1.1) and Theorem 1.2 (p. 2), clause by clause; the proof was not read, and nothing is independently reviewed in this corpus.

Depends on. Nothing in this wiki; the result is the paper's own theorem.