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We investigate a correspondence between the complexity hierarchy ofconstraint satisfaction problems and a hierarchy of logical compactnesshypotheses for finite relational structures. It seems that the harder aconstraint satisfaction problem is, the stronger the corresponding compactnesshypothesis is. At the top level, the NP-complete constraint satisfactionproblems correspond to compactness hypotheses that are equivalent to theultrafilter axiom in all the cases we have investigated. At the bottom level,the simplest constraint satisfaction problems correspond to compactnesshypotheses that are readily provable from the axioms of Zermelo and Fraenkel.

Logical compactness and constraint satisfaction problems