Canonical Vertex Partitions

Let σ be a finite relational signature and T a set of finite complete relational structures of signature σ and HT the countable homogeneous relational structure of signature σ which does not embed any of the structures in T . In the case that σ consists of at most binary relations and 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 T is determined. If 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 σ contains relational symbols of arity larger than two or that the set of bounds T is infinite.