A canonical Ramsey theorem for exactly \(m\)-coloured complete subgraphs

Abstract

Given an edge colouring of a graph with a set of \(m\) colours, we say that the graph is (exactly) \(m\)-coloured if each of the colours is used. We consider edge colourings of the complete graph on \(\mathbb{N}\) with infinitely many colours and show that either one can find an \(m\)-coloured complete subgraph for every natural number \(m\) or there exists an infinite subset \(X \subset \mathbb{N}\) coloured in one of two canonical ways: either the colouring is injective on \(X\) or there exists a distinguished vertex \(v\) in \(X\) such that \(X \setminus \lbrace v \rbrace\) is \(1\)-coloured and each edge between \(v\) and \(X \setminus \lbrace v \rbrace\) has a distinct colour (all different to the colour used on \(X \setminus \lbrace v \rbrace\)). 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 \(m\)-coloured complete subgraphs in colourings with finitely many colours.