A canonical Ramsey theorem for exactly $m$-coloured complete subgraphs

TL;DR

This paper establishes a canonical form for edge colourings of infinite complete graphs, showing that either all finite numbers of colours can be realized in subgraphs or the colouring follows one of two specific canonical patterns.

Contribution

It proves a canonical Ramsey theorem for exactly $m$-coloured subgraphs in infinite graphs, answering a question from 1999 and extending understanding of colourings with finitely many colours.

Findings

Either all finite $m$-coloured subgraphs exist or the colouring is canonical.

Identifies two canonical forms: injective colouring and a vertex with a unique colour pattern.

Provides techniques to analyze $m$-coloured subgraphs in finitely coloured graphs.

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.