Wild theories with o-minimal open core

Abstract

Let $T$ be a consistent o-minimal theory extending the theory of densely ordered groups and let $T^{\prime }$ be a consistent theory. Then there is a complete theory $T^{\ast }$ extending $T$ such that $T$ is an open core of $T^{\ast }$, but every model of $T^{\ast }$ interprets a model of $T^{\prime }$. If $T^{\prime }$ is NIP, $T^{\ast }$ 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.