For a graph and an integer , the -color Ramsey number is the least integer such that every -coloring of the edges of the complete graph contains a monochromatic copy of . Let denote the cycle on vertices. For odd cycles, Bondy and Erd\H{o}s in 1973 conjectured that for all and , . Recently, this conjecture has been verified to be true for all fixed and all sufficiently large by Jenssen and Skokan; and false for all fixed and all sufficiently large by Day and Johnson. Even cycles behave rather differently in this context. Little is known about the behavior of in general. In this talk we will present our recent results on Ramsey numbers of cycles under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. We prove that the aforementioned conjecture holds for all and all under Gallai colorings. We also completely determine the Ramsey number of even cycles under Gallai colorings.
Joint work with Dylan Bruce, Christian Bosse, Yaojun Chen and Fangfang Zhang.