In a joint work in progress with Wiesław Kubiś, Aleksandra Kwiatkowska, and Maciej Malicki we study ultrametric spaces as two-sorted structures where the distance set is a linear order that can vary from space to space. The family of all finite two-sorted ultrametric spaces with distance-carrying embeddings is a Fraïssé category, and the Fraïssé limit has a generic automorphism.

In the talk I will recall the basic Fraïssé-theoretic notions, give a motivation for switching the perspective from classical ultrametric spaces with isometric embeddings to two-sorted ultrametric spaces with dc-embeddings, and describe the generic countable space. Then we focus on (cofinal) amalgamation of partial dc-automorphisms, leading to the existence of a generic automorphism. I will sketch the proof of CAP by introducing a somewhat general strategy of splitting the problem to two parts: showing the amalgamation property of total automorphisms and showing that there are cofinally many partial automorphisms with unique totalization.