On model-theoretic tree properties

Abstract

We study model theoretic tree properties ([Formula: see text]) and their associated cardinal invariants ([Formula: see text], respectively). In particular, we obtain a quantitative refinement of Shelah’s theorem ([Formula: see text]) for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak [Formula: see text] is equivalent to [Formula: see text] (answering a question of Kim and Kim). Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed [Formula: see text].