# More on tree properties

## TL;DR

This paper explores the properties of tree properties in model theory, providing criteria for SOP$_2$ and SOP$_1$, and examining the relationship between TP$_2$ and Kim-forking, with implications for supersimplicity.

## Contribution

It introduces type-counting criteria for SOP$_2$ and SOP$_1$, and investigates the connection between TP$_2$ and Kim-forking, advancing understanding of independence notions.

## Key findings

- Type-counting criteria for SOP$_2$ and SOP$_1$ are established.
- A theory is supersimple iff there are no countably infinite Kim-forking chains.
- Relationships between TP$_2$ and Kim-forking are clarified.

## Abstract

Tree properties are introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP$_1$ or TP$_2$. In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion as it satisfies symmetry, transitivity, extension, local character, and type-amalgamation. Shelah also introduced SOP$_n$ ($n$-strong order property). Recently it is proved that in any NSOP$_1$ theory (i.e. a theory not having SOP$_1$) holding nonforking existence, Kim-forking also satisfies all the mentioned independence properties except base monotonicity (one direction of transitivity). These results are the sources of motivation for this paper. Mainly, we produce type-counting criteria for SOP$_2$ (which is equivalent to TP$_1$) and SOP$_1$. In addition, we study relationships between TP$_2$ and Kim-forking, and obtain that a theory is supersimple iff there is no countably infinite Kim-forking chain.