Amalgamation is Undecidable.
Manuel Bodirsky, Simon Knäuer, Jakub Rydval · arXiv (Cornell University) · 2021
The finite models of a universal sentence $\Phi$ are the age of a homogeneous structure if and only if $\Phi$ has the amalgamation property. We prove that the computational problem whether a given universal sentence $\Phi$ has the amalgamation property is undecidable, even if $\Phi$ is additionally Horn and the signature of $\Phi$ only contains relation symbols of arity at most three.