In this paper, we prove the following:Theorem. Let M be a rosy dependent theory and letp,pbe non-þ-forking extensions ofp∈Switha0a1; assume thatp∪pis consistent and thata0,a1start a þ-independent indiscernible sequence. Thenp∪pis a non-þ-forking extension ofp.We also provide an example to show that the result is not true without assuming NIP.

In a relational language consisting of a single relation R, we investigate pseudofiniteness of certain Hrushovski constructions obtained via predimension functions. It is notable that the arity of the relation R plays a crucial role in this context. When R is ternary, by extending the methods recently developed by Brody and Laskowski, we interpret 〈Q+,<〉 in the 〈K+,≤∗〉-generic and prove that this structure is not pseudofinite. This provides a negative answer to the question posed in an earlier work by Evans (...) and Wong. This result, in fact, unfolds another aspect of complexity of this structure, along with undecidability and the strict order property proved in the mentioned earlier works. On the other hand, when R is binary, it can be shown that the 〈K+,≤∗〉-generic is decidable and pseudofinite. (shrink)

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 $NTP_{2}$, strongness, $NSOP_{1}$, 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 $NTP_{2}$, the expansion of algebraically closed fields with generic automorphism is a simple theory. Furthermore, these theories are strong with $\operatorname {\mathrm {bdn}}(\text (...) {"}x=x\text {"})=(\aleph _0)_{-}$. (shrink)