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

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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) 102-115 14 Combinatorics Probability and Computing 23 1 https://doi.org/10.1017/S0963548313000503 Published - Jan 1 2014

### Fingerprint

Ramsey's Theorem
Subgraph
Color
Coloring
Edge Coloring
Graph in graph theory
Natural number
Injective
Complete Graph
Colouring
Resolve
Distinct
Subset
Vertex of a graph

### All Science Journal Classification (ASJC) codes

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

### Cite this

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

In: Combinatorics Probability and Computing, Vol. 23, No. 1, 01.01.2014, p. 102-115.

Research output: Contribution to journalArticle

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