Generic Expansions of Ordered Structures

Let T be a complete first order theory in a language L satisfying uniform finiteness and let TP be the model companion of T in an expanded language LP with a new unary predicate P . We show that: TP satisfies uniform finiteness; if T is an expansion of the theory of dense linear order then every cut definable in a model M of TP is definable in M L; and, if T is an expansion of the theory dense linear order all of whose models have o-minimal open core then every open set definable in a model M of TP is definable in M L.