## Abstract

In this paper, we study Ramsey-type problems for directed graphs. We first consider the k-colour oriented Ramsey number of H, denoted by r→(H,k), which is the least n for which every k-edge-coloured tournament on n vertices contains a monochromatic copy of H. We prove that r→(T,k)≤c _{k} |T| ^{k} for any oriented tree T. This is a generalisation of a similar result for directed paths by Chvátal and by Gyárfás and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor. We also consider the k-colour directed Ramsey number r↔(H,k) of H, which is defined as above, but, instead of colouring tournaments, we colour the complete directed graph of order n. Here we show that r↔(T,k)≤c _{k} |T| ^{k−1} for any oriented tree T, which is again tight up to a constant factor, and it generalises a result by Williamson and by Gyárfás and Lehel who determined the 2-colour directed Ramsey number of directed paths.

Original language | English (US) |
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Pages (from-to) | 145-177 |

Number of pages | 33 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 137 |

DOIs | |

State | Published - Jul 2019 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

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

## Keywords

- Directed graphs
- Ramsey theory
- Tournaments
- Trees