We show that there is a separable AF-algebra with the property that any separable AF-algebra is isomorphic to a quotient of it. This AF-algebra is the direct limit of a sequence of finite-dimensional C*-algebra where the connecting maps are left-invertible and the mentioned universality is a consequence of fact that this AF-algebra is the "Fraisse limit" of the category of all finite-dimensional C*-algebras and left-invertible embeddings. Fraisse theory also helps to describe the Bratteli diagram of this AF-algebra and provides conditions characterizing it up to isomorphisms. In some contexts, our universal AF-algebra can be considered as the appropriate noncommutative analog of the Cantor set.

Applying the K_0-functor to the result above, equivalently, there is a surjectively universal countable scaled (or with order-unit) dimension group. This is joint work with Wieslaw Kubis.