Abstract independence relations in neostability theory

TL;DR

This paper introduces a unified framework for understanding independence relations in neostability theory, generalizing key concepts and deriving broad results across various classes like simplicity, NTP2, and NSOP1.

Contribution

It extends the notion of witnessing in independence relations, unifies several results in neostability theory, and establishes a dichotomy between NSOP1 and BTP.

Findings

Proves equivalence between witnessing and symmetry in independence relations.

Derives implications for chain local character and the weak independence theorem.

Establishes a dichotomy between NSOP1 and BTP for most NSOP4 examples.

Abstract

We develop a framework, in the style of Adler, for interpreting the notion of "witnessing" that has appeared (usually as a variant of Kim's Lemma) in different areas of neostability theory as a binary relation between abstract independence relations. This involves extending the relativisations of Kim-independence and Conant-independence due to Mutchnik to arbitrary independence relations. After developing this framework, we show that several results from simplicity, $\text{NTP}_2$, $\text{NSOP}_1$, and beyond follow as instances of general theorems for abstract independence relations. In particular, we prove the equivalence between witnessing and symmetry and the implications from this notion to chain local character and the weak independence theorem, and recover some partial converses. Finally, we use this framework to prove a dichotomy between $\text{NSOP}_1$ and Kruckman and Ramsey's $\text{BTP}$ that applies to most known $\text{NSOP}_4$ examples in the literature.