Metrically Universal Generic Structures in Free Amalgamation Classes

We prove that each ∀1 free amalgamation class K over a finite relational language L admits a countable generic structure M isometrically embedding all countable structures in K relative to a fixed metric. We expand L by infinitely many binary predicates expressing distance, and prove that the resulting expansion of K has a model companion axiomatized by the first-order theory of M . The model companion is non-finitely axiomatizable, even over a strong form of the axiom scheme of infinity. Mathematics Subject Classification: 03C15, 05C12.