We shall describe the theory of inverse Fraisse limits of countable structures, namely, homogeneous ultrametric structures that can be characterized by a certain extension property. In order to construct them, we use the universal homogeneous endomorphism of a classical Fraisse limit, which exists when the Fraisse class satisfies a stronger variant of the amalgamation property.

As a sample application, we obtain the existence of an ultrametric random graph, namely, a universal homogeneous closed graph on the Baire space.

Joint work with Maja and Christian Pech.