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.