## Abstract

In this paper we study three-color Ramsey numbers. Let K_{i,j} denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G_{1}, G_{2} and G_{3}, if r(G _{1}, G_{2}) ≥ s(G_{3}), then r(G_{1}, G _{2}, G_{3}) ≥ (r(G_{1}, G_{2}) - 1)(χ(G_{3}) - 1) + s(G_{3}), where s(G_{3}) is the chromatic surplus of G_{3}; (ii) (k + m - 2) (n - 1) + 1 ≤ r(K _{1,k}, K_{1,m}, K_{n}) ≤ (k + m - 1)(n - 1) + 1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m ≥ k ≥ 2, there is a constant c such that r(K_{k,m}, K _{k,m}, K_{n}) ≤ c(n/log n)^{k}, and r(C _{2m}, C_{2m}, K_{n}) ≤ c(n/log n)^{m/(m-1)} for sufficiently large n.

Original language | English |
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Pages (from-to) | 249-258 |

Number of pages | 10 |

Journal | Graphs and Combinatorics |

Volume | 19 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2003 |

## Scopus Subject Areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

## User-Defined Keywords

- Monochromatic graph
- Three-color Ramsey number