Homogeneity and Homogenizability: Hard Problems for the Logic SNP

The finite models of a universal sentence Φ in a finite relational signature are the age of a homogeneous structure if and only if Φ has the amalgamation property. We prove that the computational problem whether a given universal sentence Φ has the amalgamation property is PSPACE-hard, even if Φ is additionally Horn and the signature of Φ only contains relation symbols of arity at most three. The decidability of the problem remains open.