Generalized and geometric Ramsey numbers for cycles

Let Cn denote the cycle of length n. The generalized Ramsey number of the p air (C-n, C-k), denoted by R(C-n,C-k), is the smallest positive integer R s uch that any complete graph with R vertices whose edges are coloured with t wo different colours contains either a monochromatic cycle of length n in t he first colour or a monochromatic cycle of length k in the second colour. Generalized Ramsey numbers for cycles were completely determined by Faudree -Schelp and Rosta, based on earlier works of Bondy, Erdos and Gallai. Unfor tunately, both proofs are quite involved and difficult to follow. In the pr esent paper we treat this problem in a unified, self-contained and simplifi ed way. We also extend this study to a related geometric problem, where we colour the straight-line segments determined by a finite number of points i n the plane. In this case, the monochromatic subgraphs are required to sati sfy an additional (non-crossing) geometric condition. (C) 2001 Elsevier Sci ence BN. All rights reserved.