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**1 - 2**of**2**### Stable embeddedness and NIP

, 2010

"... We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let P be P with its “induced ∅-definable structure”. The conditions are that P (or rather its theory) is “rosy”, P has NIP in T and that P is stably 1-embedded in T. This generalizes a recent result ..."

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We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let P be P with its “induced ∅-definable structure”. The conditions are that P (or rather its theory) is “rosy”, P has NIP in T and that P is stably 1-embedded in T. This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T. Our proofs make use of the theory of strict nonforking and weight in NIP theories ([3], [10]). 1 Introduction and

### MINIMAL TYPES IN STABLE BANACH SPACES

"... Abstract. 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 ℓ2 space. We conclude with a proof of the following formula ..."

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Abstract. 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 ℓ2 space. We conclude with a proof of the following formulation of Henson’s Conjecture: every model of an uncountably categorical theory expanding a Banach space is prime over a spreading sequence isometric to the standard basis of a Hilbert space.