3-colored Ramsey Numbers of Odd Cycles

Recently we determined the Ramsey Number r(C7, C7, C7) = 25. Let G = (V (G), E(G)) be an undirected finite graph without any loops or multiple edges, where V (G) denotes its vertex set and E(G) its edge set. In the following we will often consider the complete graph Kp on p vertices and the cycle Cp on p vertices. A k−coloring (F1, F2, . . . , Fk) of a graph G is a coloring of the edges of G with at most k different colors F1, . . . , Fk. The graph < Fi >= (V (G), E(Fi)) denotes the subgraph of G which consists of all vertices of G and all edges which are colored with color Fi. We say that Kp −→ (G1, G2, . . . , Gk), if in each k−coloring of Kp the subgraph < Fi > contains a graph isomorphic to Gi for at least one i with 1 ≤ i ≤ k. Now the Ramsey Number of k graphs G1, G2, . . . , Gk is defined as the minimal integer p such that Kp −→ (G1, . . . , Gk) (r(G1, . . . , Gk) := min{p|Kp −→ (G1, . . . , Gk)}). A good and detailed overview about known estimations and exact values is given in Radziszowski’s survey ’Small Ramsey Numbers’, [8]. We have a general lower bound only in the case k = 2, namely r(G,H) ≥ (χ(G)−1)(c(H)−1)+1, [3], where χ(G) denotes the chromatic number of G, and c(H) the order of the largest component ofH. This estimation is sharp ifG is isomorphic to the complete graph on n vertices and H is isomorphic to any tree on m vertices, namely Chvatal proved that r(Kn, Tm) = (n−1)(m−1)+1 [2]. Preprint submitted to Elsevier Preprint 29 April 1999 In this extended abstact we will mainly concentrate on complete graphs and cycles. The following table (taken from Radziszowski’s survey) contains the results for the classical Ramsey Numbers r(Kk, Kl) for small k and l. l k 3 4 5 6 7 8 9 10 11 12 13 14 15 3 6 9 14 18 23 28 36 40 43 46 51 52 60 59 69 66 78 73 89 4 18 25 35 41 49 61 55 84 69 115 80 149 96 191 128 238 131 291 136 349 145 417 5 43 49 58 87 80 143 95 216 116 316 141 442 153 181 193 221 237 6 102 165 109 298 122 495 153 780 167 1171 203 224 242 258 338 7 205 54