### Abstract

The Ramsey number r(D_{1},...,D_{k}) of acyclic directed graphs D_{1},...,D_{k} is defined as the largest integer r for which there exists a tournament T = (V,A) on r vertices with a k-coloring φ:A → {1,...,k} of the arc set A such that no D_{i} occurs in color i for any i ∈ {1,...,k}. We discuss recursive techniques to compute r(D_{1},...,D_{k}) in the case where there are paths and/or stars among the D_{i}. In particular, solving a problem of Bialostocki and Dierker [Congr. Numer. 47 (1985) 119-123], we prove that r(D_{1},D_{2})=r(D_{1})·r(D_{2}) holds if D_{1} is transitive and D_{2} = S_{n} is an out-going star on n vertices. Our main result is an asymptotic formula for r(D_{1},...,D_{k},S_{n}) where the digraphs D_{1},...,D_{k} are fixed arbitrarily and n → ∞.

Original language | English |
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Pages (from-to) | 75-85 |

Number of pages | 11 |

Journal | Theoretical Computer Science |

Volume | 263 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Aug 14 2001 |

### Keywords

- Ramsey number
- Tournament

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theoretical Computer Science*,

*263*(1-2), 75-85. https://doi.org/10.1016/S0304-3975(00)00232-2