Local order property in nonelementary classes

We study a local version of the order property in several frameworks, with an emphasis on frameworks where the compactness theorem fails: (1) Inside a fixed model, (2) for classes of models where the compactness theorem fails and (3) for the first order case. Appropriate localizations of the order p ropel-ty, the independence property, and the strict order property are intr oduced. We are able to generalize some of the results that were known in th e case of local stability for the first order theories, and for stability f or nonelementary classes (existence of indiscernibles, existence of average s, stability spectrum, equivalence between order and instability). In the f irst order case, we also prove the local version of Shelah's Trichotomy The orem. Finally, as an application, we give a new characterization of stable types when the ambient first order theory is simple.