Model Theory and Combinatorics of Homogeneous Metric Spaces

We develop the model theory of generalized metric spaces, in which distances between points are taken from arbitrary ordered additive structures. Our focus is on the class of structures obtained by fixing a countable, positively ordered monoid R, and considering the universal, ultrahomogeneous Urysohn space U(R), which takes distances in R. As notable examples, this class includes the Fraisse limits of rational metric spaces, graphs, and refining equivalence relations, as well as more general structures previously studied by Delhomme, Laflamme, Pouzet, and Sauer. We characterize quantifier elimination in the theory of U(R) (in a relational language) using continuity properties in R, which hold in most natural examples arising in previous literature. In the case where quantifier elimination holds, we consider classification theoretic “neostability” properties of U(R), focusing especially on when such properties are characterized by first-order statements about the monoid R. As a first example of this phenomenon, we prove that U(R) is stable if and only if it is an ultrametric space. Using a characterization of forking for complete types, we are also able to characterize simplicity in terms of “low complexity” in the monoid R. We then generalize the characterizations of stability and simplicity, in order to define the integer-valued “archimedean complexity” of an ordered monoid. We show that the position of U(R) in Shelah's hierarchy of strong order properties is precisely determined by this invariant, which gives the first class of natural examples in which the entirety of this hierarchy can be meaningfully interpreted. We also generalize previous work of Casanovas and Wagner to obtain necessary conditions for elimination of hyperimaginaries and weak elimination of imaginaries in U(R). The last two chapters shift in focus to the combinatorics of these Urysohn spaces. We generalize work of Solecki to prove that, when R is archimedean, the class of finite R-metric spaces has the Hrushovski property. We then extend this result to a larger class, which includes ultrametric spaces. Finally, we consider the asymptotic behavior of finite distance monoids, and uncover some surprising connections to other areas of algebraic and additive combinatorics.