### 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 |
---|---|

Pages (from-to) | 75-85 |

Number of pages | 11 |

Journal | Theoretical Computer Science |

Volume | 263 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2001 |

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### Keywords

- Ramsey number
- Tournament

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Theoretical Computer Science*,

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

**Ramsey numbers for tournaments.** / Manoussakis, Yannis; Tuza, Z.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 263, no. 1-2, pp. 75-85. https://doi.org/10.1016/S0304-3975(00)00232-2

}

TY - JOUR

T1 - Ramsey numbers for tournaments

AU - Manoussakis, Yannis

AU - Tuza, Z.

PY - 2001

Y1 - 2001

N2 - The Ramsey number r(D1,...,Dk) of acyclic directed graphs D1,...,Dk 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 Di occurs in color i for any i ∈ {1,...,k}. We discuss recursive techniques to compute r(D1,...,Dk) in the case where there are paths and/or stars among the Di. In particular, solving a problem of Bialostocki and Dierker [Congr. Numer. 47 (1985) 119-123], we prove that r(D1,D2)=r(D1)·r(D2) holds if D1 is transitive and D2 = Sn is an out-going star on n vertices. Our main result is an asymptotic formula for r(D1,...,Dk,Sn) where the digraphs D1,...,Dk are fixed arbitrarily and n → ∞.

AB - The Ramsey number r(D1,...,Dk) of acyclic directed graphs D1,...,Dk 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 Di occurs in color i for any i ∈ {1,...,k}. We discuss recursive techniques to compute r(D1,...,Dk) in the case where there are paths and/or stars among the Di. In particular, solving a problem of Bialostocki and Dierker [Congr. Numer. 47 (1985) 119-123], we prove that r(D1,D2)=r(D1)·r(D2) holds if D1 is transitive and D2 = Sn is an out-going star on n vertices. Our main result is an asymptotic formula for r(D1,...,Dk,Sn) where the digraphs D1,...,Dk are fixed arbitrarily and n → ∞.

KW - Ramsey number

KW - Tournament

UR - http://www.scopus.com/inward/record.url?scp=0034922350&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034922350&partnerID=8YFLogxK

U2 - 10.1016/S0304-3975(00)00232-2

DO - 10.1016/S0304-3975(00)00232-2

M3 - Article

VL - 263

SP - 75

EP - 85

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 1-2

ER -