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Claim. There is an $n_0$ such that for all odd $n_1,n_2,n_3>n_0$,

$$R(C_{n_1},C_{n_2},C_{n_3})=4\max\{n_1,n_2,n_3\}-3;$$

in particular $R_3(C_n)=4n-3$ for every odd $n>n_0$, which is the problem's inequality with equality. The lower bound $R_3(C_n)\ge4n-3$ holds for every odd $n$ by two explicit colorings of $K_{4n-4}$ (the paper's Claim 2), and the upper bound comes from a stability theorem proved by the regularity method, so $n_0$ is not made explicit. The statement is paged at Theorem 1 of the library's source card, which describes the 38-page CDAM Research Report LSE-CDAM-2008-16 whose pages and statement numbers the problem page uses.

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.

Standing. Claimed, not accepted. The site's curator, T. F. Bloom, credits Kohayakawa, Simonovits and Skokan in the problem's commentary with proving the conjecture for all sufficiently large odd $n$, on a page labeled DECIDABLE (last edited 8 February 2026), the site's state for a problem resolved up to a finite check, which rests on exactly this theorem and the even-cycle theorem of Benevides and Skokan; that label does not mark the problem settled, so the credit is recorded here and is not reviewed evidence. Nothing is refereed: the claimant's own venue is the extended abstract The 3-colored Ramsey number of odd cycles, Proceedings of GRACO2005, Electron. Notes Discrete Math. 19 (2005), 397--402, dated June 2005 by its Crossref record, the date this page is named by; it is a six-page abstract in a proceedings series, not compared with the report, and not shown to have been refereed. The complete proof is the CDAM Research Report LSE-CDAM-2008-16 linked above (file dated 22 September 2008), which is not refereed; no journal version was found. A refereed publication of the proof, or an independent reviewer's acceptance of it, would move the claim to accepted. The same statement is proved again, for every fixed number of colors, by Jenssen and Skokan (Adv. Math. 2021), recorded on its own page Jenssen and Skokan 2016; that paper describes its result as a stability-type strengthening of this paper's main result and likewise gives no effective threshold. It corroborates this page and lends it no evidence.

Read depth. Claims checked: Theorem 1, Claim 2 and Theorem 3 of the report (pp. 2--5); 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.