### Abstract

Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. We consider edge colourings of the complete graph on ℕ with infinitely many colours and show that either one can find an exactly m-coloured complete subgraph for every natural number m or there exists an infinite subset X ⊂ ℕ coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex ν in X such that X\{ν} is 1-coloured and each edge between ν and X\{ν} has a distinct colour (all different to the colour used on X\{ν}). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding exactly m-coloured complete subgraphs in colourings with finitely many colours.

Original language | English (US) |
---|---|

Pages (from-to) | 102-115 |

Number of pages | 14 |

Journal | Combinatorics Probability and Computing |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Combinatorics Probability and Computing*,

*23*(1), 102-115. https://doi.org/10.1017/S0963548313000503

}

*Combinatorics Probability and Computing*, vol. 23, no. 1, pp. 102-115. https://doi.org/10.1017/S0963548313000503

**A canonical ramsey theorem for exactly m-coloured complete subgraphs.** / Kittipassorn, Teeradej; Narayanan, Bhargav P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A canonical ramsey theorem for exactly m-coloured complete subgraphs

AU - Kittipassorn, Teeradej

AU - Narayanan, Bhargav P.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. We consider edge colourings of the complete graph on ℕ with infinitely many colours and show that either one can find an exactly m-coloured complete subgraph for every natural number m or there exists an infinite subset X ⊂ ℕ coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex ν in X such that X\{ν} is 1-coloured and each edge between ν and X\{ν} has a distinct colour (all different to the colour used on X\{ν}). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding exactly m-coloured complete subgraphs in colourings with finitely many colours.

AB - Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. We consider edge colourings of the complete graph on ℕ with infinitely many colours and show that either one can find an exactly m-coloured complete subgraph for every natural number m or there exists an infinite subset X ⊂ ℕ coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex ν in X such that X\{ν} is 1-coloured and each edge between ν and X\{ν} has a distinct colour (all different to the colour used on X\{ν}). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding exactly m-coloured complete subgraphs in colourings with finitely many colours.

UR - http://www.scopus.com/inward/record.url?scp=84889635118&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84889635118&partnerID=8YFLogxK

U2 - 10.1017/S0963548313000503

DO - 10.1017/S0963548313000503

M3 - Article

AN - SCOPUS:84889635118

VL - 23

SP - 102

EP - 115

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 1

ER -