The 3-colored Ramsey number of even cycles

Denote by R(L, L, L) the minimum integer N such that any 3-coloring of the edges of the complete graph on N vertices contains a monochromatic copy of a graph L. Bondy and Erdős conjectured that when L is the cycle Cn on n vertices, R(Cn, Cn, Cn) = 4n − 3 for every odd n > 3. Luczak proved that if n is odd, then R(Cn, Cn, Cn) = 4n + o(n), as n → ∞, and Kohayakawa, Simonovits and Skokan confirmed the Bondy-Erdős conjecture for all sufficiently large values of n. Figaj and Luczak determined an asymptotic result for the ‘complementary’ case where the cycles are even: they showed that for even n, we have R(Cn, Cn, Cn) = 2n + o(n), as n → ∞. In this paper, we prove that there exists n1 such that for every even n ≥ n1, R(Cn, Cn, Cn) = 2n.