1 research outputs found
A canonical Ramsey theorem for exactly -coloured complete subgraphs
Given an edge colouring of a graph with a set of colours, we say that the
graph is (exactly) -coloured if each of the colours is used. We consider
edge colourings of the complete graph on with infinitely many
colours and show that either one can find an -coloured complete subgraph for
every natural number or there exists an infinite subset coloured in one of two canonical ways: either the colouring is
injective on or there exists a distinguished vertex in such that is -coloured and each edge between and has a distinct colour (all different to the colour
used on ). 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 -coloured complete subgraphs in
colourings with finitely many colours.Comment: 16 pages, improved presentation, fixed misprints, Combinatorics,
Probability and Computin