Results on Colored Tree Properties

In this article, we introduce novel variations on several well-known model-theoretic tree properties (TPs), and prove several equivalences to known properties. Motivated by the study of generalized indiscernibles, we introduce the notion of the I I script upper I -tree property ( I I script upper I -TP), for an arbitrary Ramsey index structure I I script upper I . We focus attention on the colored linear order index structure c , showing that c -TP is equivalent to instability. After introducing c - TP 1 TP₁ TP Subscript 1 and c - TP 2 TP₂ TP Subscript 2 , we prove that c - TP 1 TP₁ TP Subscript 1 is equivalent to TP 1 TP₁ TP Subscript 1 , and that c - TP 2 TP₂ TP Subsc