Claims

../

1955_01_01_greenwood_gleason: Correct, but answers the site's wording (every cycle length n at least 3), not the corrected Statement (every n > 3), so it does not count toward the problem's standing. The classical value R(3,3,3) = 17 of Greenwood and Gleason (Canad. J. Math. 1955) exceeds 4n - 3 = 9 at n = 3, the triangle.

2005_06_01_kohayakawa_simonovits_skokan: Theorem 1 of Kohayakawa, Simonovits and Skokan gives a threshold beyond which R(C_{n_1}, C_{n_2}, C_{n_3}) = 4 max(n_i) - 3 for odd lengths, so the bound holds with equality for every sufficiently large odd n.

2008_09_21_benevides_skokan: Theorem 1 of Benevides and Skokan (J. Combin. Theory Ser. B 2009) gives a threshold beyond which R(C_n, C_n, C_n) = 2n for even n, so the bound 4n - 3 holds with room to spare for every sufficiently large even n.

2016_08_19_jenssen_skokan: Theorem 1.2 of Jenssen and Skokan (Adv. Math. 2021) gives the exact k-color Ramsey number of long odd cycles for every fixed k; its case k = 3 is R_3(C_n) = 4n - 3 for all large odd n, the problem's bound with equality.

---