### Abstract

In this paper, we study the generalized Ramsey number r(G_{1},..., G_{k}) where the graphs G_{1},..., G_{k} consist of complete graphs, complete bipartite graphs, paths, and cycles. Our main theorem gives the Ramsey number for the case where G_{2},..., G_{k} are fixed and G_{1} {reversed tilde equals} C_{n} or P_{n} with n sufficiently large. If among G_{2},..., G_{k} there are both complete graphs and odd cycles, the main theorem requires an additional hypothesis concerning the size of the odd cycles relative to their number. If among G_{2},..., G_{k} there are odd cycles but no complete graphs, then no additional hypothesis is necessary and complete results can be expressed in terms of a new type of Ramsey number which is introduced in this paper. For k = 3 and k = 4 we determine all necessary values of the new Ramsey number and so obtain, in particular, explicit and complete results for the cycle Ramsey numbers r(C_{n}, C_{l}, C_{k}) and r(C_{n}, C_{l}, C_{k}, C_{m}) when n is large.

Original language | English |
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Pages (from-to) | 250-264 |

Number of pages | 15 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 20 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jun 1976 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Journal of Combinatorial Theory, Series B*,

*20*(3), 250-264. https://doi.org/10.1016/0095-8956(76)90016-2