Bi-colored expansions of geometric theories

Abstract

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 $\alpha \in (0,1]$, we define a pre-dimension function ${\delta }_{\alpha }$ on the class of Bi-colored models of $T^{\forall }$ and consider the subclass $K_{\alpha }^{+}$ consisting of models with hereditary positive ${\delta }_{\alpha }$. We impose certain natural conditions on T that enable us to introduce a complete ${\Pi }_2$-theory $T_{\alpha }$ for the rich models in $K_{\alpha }^{+}$. We show how the transfer of certain model-theoretic properties, such as NIP and strong-dependence, from T to $T_{\alpha }$, depends on whether α is rational or irrational.