The 3-colored Ramsey number of odd cycles

For graphs L1, . . . ,Lk, the Ramsey number R(L1, . . . ,Lk) is the minimum integer N satisfying that for any coloring of the edges of the complete graph KN on N vertices by k colors there exists a color i for which the corresponding color class contains Li as a subgraph. In 1973, Bondy and Erd˝os conjectured that if n is odd and Cn denotes the cy- cle on n vertices, then R(Cn,Cn,Cn) = 4n − 3. In 1999, Luczak proved that R(Cn,Cn,Cn) = 4n + o(n), where o(n)/n ! 0 as n ! 1. In this paper we strengthen Luczak’s result and verify this conjecture for n sufficiently large.