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 C n on n vertices, R ( C n , C n , C n ) = 4 n − 3 for every odd n > 3 . Łuczak proved that if n is odd, then R ( C n , C n , C n ) = 4 n + o ( n ) , as n → ∞ , and Kohayakawa, Simonovits and Skokan confirmed the Bondy–Erdős conjecture for all sufficiently large values of n. and Łuczak determined an asymptotic result for the ‘complementary’ case where the cycles are even: they showed that for even n, we have R ( C n , C n , C n ) = 2 n + o ( n ) , as n → ∞ . In this paper, we prove that there exists n 1 such that for every even n ⩾ n 1 , R ( C n , C n , C n ) = 2 n .