# Fra\"iss\'e Limits for Relational Metric Structures

## TL;DR

This paper strengthens the theory of Fra"issé limits for relational metric structures, providing conditions for exact ultrahomogeneity and applying these results to stochastic processes and diversities.

## Contribution

It introduces a new condition ensuring exact ultrahomogeneity in Fra"issé limits for relational metric structures, extending previous approximate results.

## Key findings

- Established a general condition for exact ultrahomogeneity.
- Applied the condition to stochastic processes and diversities.
- Extended Fra"issé limit theory to new classes of metric structures.

## Abstract

The general theory developed by Ben Yaacov for metric structures provides Fra\"iss\'e limits which are approximately ultrahomogeneous. We show here that this result can be strengthened in the case of relational metric structures. We give an extra condition that guarantees exact ultrahomogenous limits. The condition is quite general. We apply it to stochastic processes, the class of diversities, and its subclass of $L_1$ diversities.