Submission date: 3 May 2015.

Abstract:

We study model theoretic tree properties (TP, TP_1, TP_2) and their associated cardinal invariants (κ_{cdt}, κ_{sct}, 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.

Mathematics Subject Classification: 03C45, 05C55, 05C05

Keywords and phrases:

Full text arXiv 1505.00454: pdf, ps.