On the multi-colored Ramsey numbers of cycles

For a graph L and an integer k ≥ 2, Rk(L) denotes the smallest integer N for which for any edge-colouring of the complete graph KN by k colours there exists a colour i for which the corresponding colour class contains L as a subgraph. Bondy and Erdős conjectured that for an odd cycle Cn on n vertices, Rk(Cn) = 2 k−1(n− 1) + 1 for n > 3. They proved the case when k = 2 and also provided an upper bound Rk(Cn) ≤ (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, Rk(Cn) ≤ k2n + o(n), as n→∞. When n is even, Yongqi, Yuansheng, Feng, and Bingxi gave a construction, showing that Rk(Cn) ≥ (k − 1)n − 2k + 4. Here we prove that if n is even, then Rk(Cn) ≤ kn + o(n), as n→∞. ∗Faculty of Mathematics and Computer Science, Adam Mickiewicz University, 61-614 Poznań, Poland, e-mail: [email protected] †The author was partially supported by the Foundation for Polish Science. ‡Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences H-1053 Budapest, Reáltanoda u. 13-15., Hungary, e-mail: [email protected] §The author was partially supported by the Hungarian National Science Foundation grants OTKA T 026069, T 038210, T 0234702, and T 69062. ¶Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom, e-mail: [email protected]