Results on Colored Tree Properties

TL;DR

This paper explores new variations of model-theoretic tree properties, establishing equivalences and a dichotomy theorem, and introduces the $ ext{c}$-tree property related to instability and indiscernibles.

Contribution

It introduces the $ ext{c}$-tree property and its variants, proving their equivalences to known properties and establishing a dichotomy theorem for these properties.

Findings

$ ext{c}$-TP is equivalent to instability.

$ ext{c}$-$ ext{TP}_i$ is equivalent to $ ext{TP}_i$.

$ ext{c}$-$ ext{TP}_{ii}$ is equivalent to IP.

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.