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ESSENTIALLY INFINITE COLOURINGS OF GRAPHS

Published online by Cambridge University Press:  01 June 2000

B. BOLLOBÁS ,
Y. KOHAYAKAWA  and
R. H. SCHELP
B. BOLLOBÁS
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA Current address: Trinity College, Cambridge CB2 1TQ; bollobas@msci.memphis.edu
Y. KOHAYAKAWA
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-900 São PauloBrazil; yoshi@ime.usp.br
R. H. SCHELP
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA; schelpr@msci.memphis.edu
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Abstract

The classical canonical Ramsey theorem of Erdős and Rado states that, for any integer q [ges ] 1, any edge colouring of a large enough complete graph contains one of three canonically coloured complete subgraphs of order q. Of these canonical subgraphs, one is coloured monochromatically while each of the other two has its edge set coloured with many different colours. The paper presents a condition on colourings that, roughly speaking, requires them to make effective use of many colours (‘essential infiniteness’); this condition is then shown to imply that the colourings in question must contain large refinements of one of two ‘unavoidable’ colourings that are rich in colours. As it turns out, one of these unavoidable colourings is one of the canonical colourings of Erdős and Rado, and the other is a ‘bipartite variant’ of this colouring.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 61 , Issue 3 , June 2000 , pp. 658 - 670
Copyright
The London Mathematical Society 2000

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