Generic Expansions of Geometric Theories

As a continuation of ideas initiated in [19], we study bi-colored (generic) expansions of geometric theories in the style of the Fraïssé–Hrushovski construction method. Here we examine that the properties NTP₂ , strongness, NSOP₁ , and simplicity can be transferred to the expansions. As a consequence, while the corresponding bi-colored expansion of a red non-principal ultraproduct of p -adic fields is NTP₂ , the expansion of algebraically closed fields with generic automorphism is a simple theory. Furthermore, these theories are strong with bdn("x=x")=( ₀)₋ .