Prelegent: Ziemowit Kostana

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. We prove similar results for some other classes of models, for instance graphs. In certain cases we give a (consistent) classification of constructed models.