On the multi‐colored Ramsey numbers of cycles

For a graph L and an integer k ≥2, R k ( L ) denotes the smallest integer N for which for any edge‐coloring of the complete graph K N by k colors there exists a color i for which the corresponding color class contains L as a subgraph. Bondy and Erdos conjectured that, for an odd cycle C n on n vertices,They proved the case when k = 2 and also provided an upper bound R k ( C n )≤( k + 2)! n . Recently, this conjecture has been verified for k = 3 if n is large. In this note, we prove that for every integer k ≥4,When n is even, Sun Yongqi, Yang Yuansheng, Xu Feng, and Li Bingxi gave a construction, showing that R k ( C n )≥( k −1) n −2 k + 4. Here we prove that if n is even, then© 2011 Wiley Periodicals, Inc. J Graph Theory 69: 169‐175, 2012