We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies severa
We prove existence of wide types in a continuous theory expanding a Banach space, and density of minimal wide types among stable types in such a theory. We show that every minimal wide stable type is generically isometric to an l_2 space. We conclude
with a proof of the following formulation of Hensons Conjecture: every model of an uncountably categorical theory expanding a Banach space is prime over a spreading model, isometric to the standard basis of a Hilbert space.
We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular it follows from our investigation that in resilient theories strict non-forking is symmetric. Base
d on this study, we develop notions of weight which characterize NTP2, dependence and strong dependence. Many of our proofs rely on careful analysis of sequences that witness dividing. We prove simple characterizations of such sequences in resilient theories, as well as of Morley sequences which are witnesses. As a by-product we obtain information on types co-dominated by generically stable types in dependent theories. For example, we prove that every Morley sequence in such a type is a witness.
We characterize nonforking (Morley) sequences in dependent theories in terms of a generalization of Poizats special sequences and show that average types of Morley sequences are stationary over their domains. We characterize generically stable types
in terms of the structure of the eventual type. We then study basic properties of strict Morley sequences, based on Shelahs notion of strict nonforking. In particular we prove Kims lemma for such sequences, and a weak version of local character.
We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in descriptive set theoretic (topological) sense and in measure theoretic sense.
In particular, it gives a new perspective on Vershiks theorems on genericity and randomness of Urysohns space among separable metric spaces.
We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent (NIP) theories.
