Bi-colored Expansions of Geometric Theories
This talk is concerned with Bi-colored expansions of geometric theories in the light of the Fraisse-Hrushovski construction method. Substructures of models of a geometric $T$ are expanded by a color predicate p, where the dimension function associated with the pre-geometry of the $T$-algebraic closure operator together with a real number $0 < \alpha < 1$ give rise to a pre-dimension function $\delta_{\alpha}$. Imposing certain natural conditions on $T$, enables us to introduce a complete axiomatization $T_{\alpha}$ for the class of rich structures. It follows that if T is a dependent theory (NIP) then so is $T_{\alpha}$. We further prove that whenever $\alpha$ is rational the strong dependence transfers to $T_{\alpha}$. We are further able to show that if T defines a linear order and $\alpha$ is irrational then $T_{\alpha}$ is not strongly dependent. This is a joint work with S.Jalili and M.Khani