On model-theoretic tree properties

Abstract

We study model theoretic tree properties ($\text{TP},{\text{TP}}_1,{\text{TP}}_2$) and their associated cardinal invariants (${\kappa }_{\text{cdt}},{\kappa }_{\text{sct}},{\kappa }_{\text{inp}}$, respectively). In particular, we obtain a quantitative refinement of Shelah's theorem ($\text{TP}\Rightarrow {\text{TP}}_1\vee {\text{TP}}_2$) for countable theories, show that ${\text{TP}}_1$ is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak $k-{\text{TP}}_1$ is equivalent to ${\text{TP}}_1$ (answering a question of Kim and Kim). Besides, we give a characterization of ${\text{NSOP}}_1$ via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed ${\text{NSOP}}_1$.

v.2: Proofs of Lemma 5.2 and Prop 4.9 were clarified, Prop 6.16 - corrected; minor presentation improvements; accepted to the Journal of Mathematical Logic