Ramsey theory (2nd ed.)
Ramsey problems and their connection to Tura´n-type extremal problems
Journal of Graph Theory
Ramsey-type results for geometric graphs
Proceedings of the twelfth annual symposium on Computational geometry
Ramsey-type results for geometric graphs. II
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Discrete Mathematics
The Ramsey number for a triple of long even cycles
Journal of Combinatorial Theory Series B
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Let Cn denote the cycle of length n. The generalized Ramsey number of the pair (Cn , Ck) denoted by; R(Cn, Ck)is the smallest positive integer Rsuch that any complete graph with R vertices whose edges are coloured with two di/erent colours contains either a monochromatic cycle of length n in the 1rst 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, Erd&ohuml;s and Gallai. Unfortunately, both proofs are quite involved and di9cult to follow. In the present paper we treat this problem in a uni1ed, self-contained and simpli1ed way. We also extend this study to a related geometric problem, where we colour the straight-line segments determined by a 1nite number of points in the plane. In this case, the monochromatic subgraphs are required to satisfy an additional (non-crossing) geometric condition.