Canonical Vertex Partitions

Let $\sigma$ be a finite relational signature, let $\mathcal T$ be a set of finite complete relational structures of signature $\sigma$, and let ${\rm H}_{\mathcal T}$ be the countable homogeneous relational structure of signature $\sigma$ which does not embed any of the structures in $\mathcal T$.When $\sigma$ consists of at most binary relations and $\mathcal T$ is finite, the vertex partition behaviour of ${\rm H}_{\mathcal T}$ 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 when $\sigma$ contains relational symbols of arity larger than two or when the set of bounds $\mathcal{T}$ is infinite.