Bi-Colored Expansions of Geometric Theories

Abstract

This paper concerns the study of Bi-colored expansions of geometric theories in the light of the Fra\"{i}ss\'{e}-Hrushovski construction method. Substructures of models of a geometric theory $T$ are expanded by a color predicate $p$, and the dimension function associated with the pre-geometry of the $T$-algebraic closure operator together with a real number $0<\alpha\leqslant 1$ is used to define a pre-dimension function $\delta_{\alpha}$. The pair $(\mathcal{K}_{\alpha}^{+},\leqslant_{\alpha})$ consisting of all such expansions with a hereditary positive pre-dimension along with the notion of substructure $\leqslant_{\alpha}$ associated to $\delta_{\alpha}$ is then used as a natural setting for the study of generic bi-colored expansions in the style of Fra\"{i}ss\'{e}-Hrushovski construction. Imposing certain natural conditions on $T$, enables us to introduce a complete axiomatization $\mathbb{T}_{\alpha}$ for the class of rich structures in this class. We will show that if $T$ is a dependent theory (NIP) then so is $\mathbb{T}_{\alpha}$. We further prove that whenever $\alpha$ is rational the strong dependence transfers to $\mathbb{T}_{\alpha}$. We conclude by showing that if $T$ defines a linear order and $\alpha$ is irrational then $\mathbb{T}_{\alpha}$ is not strongly dependent.