Invariants Related to the Tree Property

TL;DR

This paper explores invariants related to the tree property in model theory, constructing theories with specific invariant relations and analyzing their impact on model saturation, thereby addressing questions posed by Shelah.

Contribution

It introduces new theories with particular relationships among Shelah's invariants and investigates their effects on saturation decay in ultrapowers, advancing understanding of the tree property.

Findings

Constructed theories with $ abla_{cdt}(T) > abla_{sct}(T) + abla_{inp}(T)$

Analyzed how invariants influence saturation decay in ultrapowers

Answered open questions of Shelah regarding these invariants.

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.