A canonical ramsey theorem for exactly m-coloured complete subgraphs

Teeradej Kittipassorn, Bhargav P. Narayanan

Research output: Contribution to journalArticle

2 Scopus citations

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 languageEnglish (US)
Pages (from-to)102-115
Number of pages14
JournalCombinatorics Probability and Computing
Volume23
Issue number1
DOIs
StatePublished - Jan 1 2014
Externally publishedYes

All Science Journal Classification (ASJC) codes

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

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