### 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) |
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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 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

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

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## Cite this

*Combinatorics Probability and Computing*,

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