University of Calgary

# Canonical vertex partitions

## Abstract

Let $\sigma$ be a finite relational signature and $\mathcal T$ a set of finite complete relational structures of signature $\sigma$ and $\Ha_{\mathcal T}$ the countable homogeneous relational structure of signature $\sigma$ which does not embed any of the structures in $\mathcal T$. In the case that $\sigma$ consists of at most binary relations and $\mathcal T$ is finite the vertex partition behaviour of $\Ht$ is completely analysed; in the sense that it is shown that a canonical partition exists and the size of this partition in terms of the structures in $\mathcal{T}$ is determined. If $\mathcal{T}$ is infinite some results are obtained but a complete analysis is still missing. Some general results are presented which are intended to be used in further investigations in case that $\sigma$ contains relational symbols of arity larger than two or that the set of bounds $\mathcal{T}$ is infinite.