University of Calgary

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.

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