Taking model-complete cores

A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; the core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, they always exist for $\omega$-categorical theories. We show that many model-theoretic properties, such as stability, NIP, simplicity, and NSOP, are preserved by moving to the core companion of a theory. On the other hand, we show that the classes of theories of structures interpretable over $({\mathbb N};=)$ and over $({\mathbb Q};<)$ are both not closed under taking core companions. The first class is contained in the class of theories of $\omega$-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80's. We conjecture the two classes to be equal.