Bi-colored expansions of geometric theories

This paper concerns the study of expansions of models of a geometric theory T by a color predicate p, within the framework of the Fraïssé-Hrushovski construction method. For each α∈(0,1], we define a pre-dimension function δα on the class of Bi-colored models of T∀ and consider the subclass Kα+ consisting of models with hereditary positive δα. We impose certain natural conditions on T that enable us to introduce a complete Π2-theory Tα for the rich models in Kα+. We show how the transfer of certain model-theoretic properties, such as NIP and strong-dependence, from T to Tα, depends on whether α is rational or irrational.