## 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. The question of finding exactly m-coloured complete subgraphs was first considered by Erickson in 1994; in 1999, Stacey and Weidl partially settled a conjecture made by Erickson and raised some further questions. In this paper, we shall study, for a colouring of the edges of the complete graph on N with exactly k colours, how small the set of natural numbers m for which there exists an exactly m-coloured complete infinite subgraph can be. We prove that this set must have size at least 2k; this bound is tight for infinitely many values of k. We also obtain a version of this result for colourings that use infinitely many colours.

Original language | English (US) |
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Pages (from-to) | 163-173 |

Number of pages | 11 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 106 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

## Keywords

- Exactly m-coloured graphs
- Ramsey theory