Characterizations of monadic NIP

We give several characterizations of when a complete first-order theory T T is monadically NIP, i.e. expansions by arbitrary unary predicates do not have the independence property. The central characterization condition on finite satisfiability types. Other include decompositions models, behavior indiscernibles, and forbidden configuration. As an application, we prove non-structure results for hereditary classes substructures non-monadically NIP models that eliminate quantifiers.