Essentially infinite colourings of graphs


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.

Publication Title

Journal of the London Mathematical Society