Results on Colored Tree Properties

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 $\text{calI}$-tree property ($\text{calI}$-TP), for an arbitrary Ramsey index structure $\text{calI}$. We focus attention on the colored linear order index structure $\mathbf{\text{c}}$, showing that $\mathbf{\text{c}}$-TP is equivalent to instability. After introducing $\mathbf{\text{c}}$-$\text{TPi}$ and $\mathbf{\text{c}}$-$\text{TPii}$, we prove that $\mathbf{\text{c}}$-$\text{TPi}$ is equivalent to $\text{TPi}$, and that $\mathbf{\text{c}}$-$\text{TPii}$ is equivalent to IP. We see that these three tree properties give a dichotomy theorem, just as with TP, $\text{TPi}$, and $\text{TPii}$. Along the way, we observe that appropriately generalized tree index structures ${\text{calI}}^{<\omega }$ are Ramsey, allowing for the use of generalized tree indiscernibles.