  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.
