### Abstract

A class Γ of graphs is vertex Ramsey if for all H ∈ Γ there exists G ∈ Γ such that for all partitions of the vertices of G into two parts, one of the parts contains an induced copy of H. Forb (T,K) is the class of graphs that induce neither T nor K. Let T(k,r) be the tree with radius r such that each nonleaf is adjacent to k vertices farther from the root than itself. Gyárfás conjectured that for all trees T and cliques K, there exists an integer b such that for all G in Forb (T,K), the chromatic number of G is at most b. Gyárfás' conjecture implies a weaker conjecture of Sauer that for all trees T and cliques K, Forb (T,K) is not vertex Ramsey. We use techniques developed for attacking Gyárfás' conjecture to prove that for all q, r and sufficiently large k, Forb (T(k,r),K_{q}) is not vertex Ramsey.

Original language | English (US) |
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Pages (from-to) | 493-504 |

Number of pages | 12 |

Journal | Combinatorica |

Volume | 16 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1996 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*16*(4), 493-504. https://doi.org/10.1007/BF01271268