Invariants Related to the Tree Property

Abstract

We consider global analogues of model-theoretic tree properties. The main objects of study are the invariants related to Shelah's tree property $\kappa_{\text{cdt}}(T)$, $\kappa_{\text{sct}}(T)$, and $\kappa_{\text{inp}}(T)$ and the relations that obtain between them. From strong colorings, we construct theories $T$ with $\kappa_{\text{cdt}}(T) > \kappa_{\text{sct}}(T) + \kappa_{\text{inp}}(T)$. We show that these invariants have distinct structural consequences, by investigating the decay of saturation in ultrapowers of models of $T$, where $T$ is some theory with $\kappa_{\text{cdt}}(T)$, $\kappa_{\text{sct}}(T)$, or $\kappa_{\text{inp}}(T)$ large and bounded. This answers some questions of Shelah.