We study model theoretic tree properties (TP, TP 1 , TP 2 ) and their associated cardinal invariants (κ cdt , κsct, κ inp , respectively). In particular, we obtain a quantitative refinement of Shelah's theorem (TP ⇒ TP 1 ∨ TP 2 ) for countable theories, show that TP 1 is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak k − TP 1 is equivalent to TP 1 (answering a question of Kim and Kim). Besides, we give a characterization of NSOP 1 via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed NSOP 1 . arXiv:1505.00454v2 [math.LO]

We study NSOP _1 theories. We define Kim-independence , which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim’s lemma, local character, symmetry, and an independence theorem, and that moreover these properties individually characterize NSOP _1 theories. We describe Kim-independence in several concrete theories and observe that it corresponds to previously studied notions of independence in Frobenius fields and vector spaces with a generic bilinear form.

We study expansions of NSOP 1 theories that preserve NSOP 1 . We prove that if T is a model complete NSOP 1 theory eliminating the quantifier ∃ ∞ , then the generic expansion of T by arbitrary constant, function, and relation symbols is still NSOP 1 . We give a detailed analysis of the special case of the theory of the generic L-structure, the model companion of the empty theory in an arbitrary language L. Under the same hypotheses, we show that T may be generically expanded to an NSOP 1 theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in NSOP 1 theories, adding instances of algebraic independence to their conclusions.Date: September 18, 2018.

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