Wild theories with o-minimal open core

TL;DR

The paper constructs expansions of o-minimal theories that preserve open core properties while interpreting arbitrary theories, including NIP theories, leading to examples of NIP structures with no distal expansion.

Contribution

It introduces a method to extend o-minimal theories to interpret any given theory, maintaining NIP when applicable, and provides examples of complex NIP structures.

Findings

Existence of a complete theory extending a given o-minimal theory with open core properties.

Construction of NIP expansions of the real field with no distal expansion.

Demonstration that such expansions can interpret arbitrary theories, including non-distal NIP theories.

Abstract

Let $T$ be a consistent o-minimal theory extending the theory of densely ordered groups and let $T'$ be a consistent theory. Then there is a complete theory $T^*$ extending $T$ such that $T$ is an open core of $T^*$, but every model of $T^*$ interprets a model of $T'$. If $T'$ is NIP, $T^*$ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.