Results on Colored Tree Properties

Abstract

In this paper, we introduce novel variations on several well-known model-theoretic tree properties, and prove several equivalences to known properties. Motivated by the study of generalized indiscernibles, we introduce the notion of the $\calI$-tree property ($\calI$-TP), for an arbitrary Ramsey index structure $\calI$. We focus attention on the colored linear order index structure \textbf{c}, showing that \textbf{c}-TP is equivalent to instability. After introducing \textbf{c}-$\TPi$ and \textbf{c}-$\TPii$, we prove that \textbf{c}-$\TPi$ is equivalent to $\TPi$, and that \textbf{c}-$\TPii$ is equivalent to IP. We see that these three tree properties give a dichotomy theorem, just as with TP, $\TPi$, and $\TPii$. Along the way, we observe that appropriately generalized tree index structures $\calI^{<\omega}$ are Ramsey, allowing for the use of generalized tree indiscernibles.