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Claim. There is an integer $n_1$ such that for every even $n>n_1$,

$$R(C_n,C_n,C_n)=2n.$$

Since $2n\le4n-3$ for $n\ge2$, the problem's inequality holds for every even $n>n_1$ with a margin of $2n-3$; the bound $4n-3$ is sharp only for odd $n$. The lower bound $R(C_n,C_n,C_n)>2n-1$ holds for every even $n\ge4$ by an explicit coloring of $K_{2n-1}$ (the paper's Lemma 2); the upper bound uses the regularity method and $n_1$ is not made explicit. The statement is paged at Theorem 1 of the library's source card, which describes the 22-page CDAM Research Report LSE-CDAM-2008-17, dated 21 September 2008, the date this page is named by.

Covers. The inequality $R_3(C_n)\le4n-3$ for all sufficiently large even $n$, with the exact value $2n$. Not covered: odd $n$ (the page Kohayakawa, Simonovits and Skokan 2005), and the even $n$ below the unnamed threshold.

Acceptance. Refereed: The 3-colored Ramsey number of even cycles, J. Combin. Theory Ser. B 99 (2009), no. 4, 690--708, in the July 2009 issue (the Crossref record). The CDAM Research Report LSE-CDAM-2008-17, linked above, was not compared with the journal text, so locators are the report's. The site's curator, T. F. Bloom, credits Benevides and Skokan in the problem's commentary with the value $R_3(C_n)=2n$ for all sufficiently large even $n$, on a page labeled DECIDABLE (last edited 8 February 2026, accessed 2026-09-17), the site's state for a problem resolved up to a finite check, which rests on exactly this theorem and the odd-cycle theorem of Kohayakawa, Simonovits and Skokan; that label does not mark the problem settled, so the credit is recorded here and is not reviewed evidence.

Read depth. Claims checked: Theorem 1 and Lemma 2 (report pp. 2--3); 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.