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Journal of the London Mathematical Society 2000 61(3):658-670; doi:10.1112/S0024610700008796
© 2000 by London Mathematical Society
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© The London Mathematical Society

Essentially Infinite Colourings of Graphs

B. Bollobás, Y. Kohayakawa and R. H. Schelp

Department of Mathematical Sciences, University of Memphis Memphis, TN 38152, USA
Instituto de Matemática e Estatística, Universidade de São Paulo Rua do Matão 1010, 05508-900 São PauloBrazil yoshi{at}ime.usp.br
Department of Mathematical Sciences, University of Memphis Memphis, TN 38152, USA, schelpr{at}msci.memphis.edu

Received 5 May 1998. Revision received 12 February 1999.

The classical canonical Ramsey theorem of Erdos and Rado states that, for any integer q ≥ 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 Erdos and Rado, and the other is a ‘bipartite variant’ of this colouring.

Trinity College, Cambridge CB2 1TQ; bollobas{at}msci.memphis.edu


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