Constraint Satisfaction with Countable Homogeneous Templates

For a fixed countable homogeneous relational structure Γ we study the computational problem whether a given finite structure of the same signature homomorphically maps to Γ. This problem is known as the constraint satisfaction problem CSP(Γ) for Γ and was intensively studied for finite Γ. We show that – as in the case of finite Γ – the computational complexity of CSP(Γ) for countable homogeneous Γ is determinded by the clone of polymorphisms of Γ. To this end we prove the following theorem which is of independent interest: The primitive positive definable relations over an ω-categorical structure Γ are precisely the relations that are invariant under the polymorphisms of Γ.

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