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Properly Edge-Coloured Subgraphs in Colourings of Bounded Degree

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Abstract

The smallest n such that every colouring of the edges of K n must contain a monochromatic star K 1,s+1 or a properly edge-coloured K t is denoted by f (s, t). Its existence is guaranteed by the Erdős–Rado Canonical Ramsey theorem and its value for large t was discussed by Alon, Jiang, Miller and Pritikin (Random Struct. Algorithms 23:409–433, 2003). In this note we primarily consider small values of t. We give the exact value of f (s, 3) for all s ≥ 1 and the exact value of f (2, 4), as well as reducing the known upper bounds for f (s, 4) and f (s, t) in general.

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Authors and Affiliations

  1. Institutionen för matematik och matematisk Statistik, Umeå Universitet, Umeå, Sweden

    Klas Markström

  2. DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK

    Andrew Thomason

  3. Department Mathematik, Universität Hamburg, Bundesstr. 55, 20146, Hamburg, Germany

    Peter Wagner

Authors
  1. Klas Markström
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  2. Andrew Thomason
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  3. Peter Wagner
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Corresponding author

Correspondence to Andrew Thomason.

Additional information

P. Wagner’s research funded by Trinity College, University of Cambridge.

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Markström, K., Thomason, A. & Wagner, P. Properly Edge-Coloured Subgraphs in Colourings of Bounded Degree. Graphs and Combinatorics 27, 243–249 (2011). https://doi.org/10.1007/s00373-010-0970-5

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  • DOI: https://doi.org/10.1007/s00373-010-0970-5

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