By classical results of Poizat, the theory of beautiful pairs of models of a stable theory $T$ is "meaningful" precisely when the set of all definable types in $T$ is strict pro-definable, which is the case if and only if T is nfcp.

We transfer the notion of beautiful pairs to unstable theories and study them in particular in henselian valued fields, establishing Ax-Kochen-Ershov principles for various questions in this context. Using this, we show that the theory of beautiful pairs of models of ACVF is "meaningful" and infer the strict pro-definability of various spaces of definable types in ACVF, e.g., the model theoretic analogue of the Huber analytification of an algebraic variety.

This is joint work with Pablo Cubides Kovacsics and Jinhe Ye.