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 from \cite{shelah1990classification}.

中文翻译：

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与树属性相关的不变量

我们考虑模型理论树属性的全局类似物。主要研究对象是与 Shelah 树属性 $\kappa_{\text{cdt}}(T)$、$\kappa_{\text{sct}}(T)$ 和 $\kappa_{\text {inp}}(T)$ 以及它们之间获得的关系。从强着色，我们用 $\kappa_{\text{cdt}}(T) > \kappa_{\text{sct}}(T) + \kappa_{\text{inp}}(T) 构建理论 $T$ $. 我们通过研究 $T$ 模型的超幂的饱和度衰减，表明这些不变量具有明显的结构后果，其中 $T$ 是一些理论，其中 $\kappa_{\text{cdt}}(T)$, $\ kappa_{\text{sct}}(T)$ 或 $\kappa_{\text{inp}}(T)$ 大且有界。这回答了 \cite{shelah1990classification} 中的一些问题。