SAMS Subject Classification Number: 6

Let \(F\) and \(H\) be two bipartite graphs with Ramsey number \(R(F,H)\). Further, let \(G\) be a complete \(k\)-partite graph \(K(n_1,n_2,\ldots,n_k )\) of order \(n = \sum_{i=1}^{k} n_i\) with \(n_i\in\{\lceil \frac{n}{k} \rceil, \lfloor \frac{n}{k} \rfloor\}\) for \(i = 1,\ldots,k\) and \(k = 2,\ldots,R(F,H)\). The \(k\)-Ramsey number \(R_k(F,H)\) is then defined as the smallest positive integer \(n\) such that for any red-blue coloring of the edges of \(G\) there is a subgraph of \(G\) isomorphic to \(F\) whose edges are all colored red, or a subgraph of \(G\) isomorphic to \(H\) whose edges are all colored blue. Results on \(R_k(F,H)\) is known for two bipartite graphs \(F\) and \(H\) only. In this talk we investigate the \(k\)-Ramsey number of two cycles which are not both bipartite.