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On theories without the tree property of the second kind

Abstract : This thesis in pure model theory presents the first systematic study of the class of NTP2 theories introduced by Shelah, with a special accent on the NIP case. In the first and second chapters we develop the theory of forking over extension bases (e.g. we prove existence of universal Morley sequences, equality of forking and dividing, an independence theorem and equality of Lascar type and compact type) thus making it possible to view the results of Kim and Pillay on simple theories as a special case, but also providing a missing counterpart for the case of NIP theories. This answers questions of Adler, Hrushovski and Pillay. In the third chapter we develop the basics of the theory of burden (a generalization of the weight calculus), in particular we show that it is submultiplicative, answering a question of Shelah. We then study simple and NIP types in NTP2 theories: we prove that simple types are co-simple, characterized by the co-independence theorem, and forking between realizations of a simple type and arbitrary elements satisfies full symmetry; we show that a type is NIP if and only if all of its extensions have only boundedly many global non-forking extensions. We also prove an Ax-Kochen type preservation of NTP2, thus showing that e.g. any ultraproduct of p-adics is NTP2. We go on to study the special case of NIP theories. In Chapter 4 we introduce honest definitions and using them give a new proof of the Shelah expansion theorem and a general criterion for dependence of an elementary pair. As an application we show that naming a small indiscernible sequence preserves NIP. In Chapter 5, we combine honest definitions with some deeper combinatorial results from the Vapnik-Chervonenkis theory to deduce that in NIP theories, types over finite sets are uniformly definable. This confirms a conjecture of Laskowski for NIP theories. Besides, we give a new sufficient condition for a theory of a pair to eliminate quantifiers down to the predicate (in particular answering a question of Baldwin and Benedikt about naming an indiscernible sequence) and some examples concerning definability of 1-types vs definability of n-types over models. The last chapter is devoted to the study of non-forking spectra. To a countable first-order theory we associate its non-forking spectrum — a function of two cardinals kappa and lambda giving the supremum of the possible number of types over a model of size lambda that do not fork over a sub-model of size kappa. This is a natural generalization of the stability function of a theory. We make progress towards classifying the non-forking spectra. Besides, we answer a question of Keisler regarding the number of cuts a linear order may have. Namely, we show that it is possible that κ < (ded κ)ω
Keywords : Mathematics
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Artem Chernikov. On theories without the tree property of the second kind. Group Theory [math.GR]. Université Claude Bernard - Lyon I, 2012. English. ⟨NNT : 2012LYO10170⟩. ⟨tel-02628527⟩

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