What would the rational Urysohn space and the random graph look like if they were uncountable?

Abstract

Building on the work of Avraham, Rubin, and Shelah, we aim to build a variant of the Fra\"iss\'e theory for uncountable models built from finite submodels. With this aim, we generalize the notion of an increasing set of reals to other structures. As an application, we prove that the following is consistent: there exists an uncountable, separable metric space $X$ with rational distances, such that every uncountable partial 1-1 function from $X$ to $X$ is an isometry on an uncountable subset. We aim for a general theory of structures with this kind of properties. This includes results about the automorphism groups, and partial classification results.