  We apply the technology developed in the 80s by Avraham, Rubin, and Shelah,
to prove that the following is consistent with ZFC: 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. This space contains a dense copy of the rational Urysohn
space, and is homogeneous with respect to finite subspaces. We prove similar
results for some other classes of structures, and in some cases show that the
models we obtain are, in some sense, canonical.
