Bi-Colored Expansions of Geometric Theories

This paper concerns the study of Bi-colored expansions of geometric theories in the light of the Fraïssé-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 $0<α\leqslant 1$ is used to define a pre-dimension function $δ_α$. The pair $(\mathcal{K}_α^{+},\leqslant_α)$ consisting of all such expansions with a hereditary positive pre-dimension along with the notion of substructure $\leqslant_α$ associated to $δ_α$ is then used as a natural setting for the study of generic bi-colored expansions in the style of Fraïssé-Hrushovski construction. Imposing certain natural conditions on $T$, enables us to introduce a complete axiomatization $\mathbb{T}_α$ for the class of rich structures in this class. We will show that if $T$ is a dependent theory (NIP) then so is $\mathbb{T}_α$. We further prove that whenever $α$ is rational the strong dependence transfers to $\mathbb{T}_α$. We conclude by showing that if $T$ defines a linear order and $α$ is irrational then $\mathbb{T}_α$ is not strongly dependent.

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