A New Kim's Lemma

TL;DR

This paper introduces a new variant of Kim's Lemma that unifies existing variants for NTP2 and NSOP1 theories, expanding the understanding of independence in simple theories.

Contribution

It presents a novel Kim's Lemma variant that generalizes previous versions for NTP2 and NSOP1 theories, with exploration of its applications and implications.

Findings

The new Kim's Lemma variant holds in certain classes of theories.

It connects syntactic properties with independence notions.

Examples and non-examples illustrate the lemma's scope.

Abstract

Kim's Lemma is a key ingredient in the theory of forking independence in simple theories. It asserts that if a formula divides, then it divides along every Morley sequence in type of the parameters. Variants of Kim's Lemma have formed the core of the theories of independence in two orthogonal generalizations of simplicity - namely, the classes of NTP2 and NSOP1 theories. We introduce a new variant of Kim's Lemma that simultaneously generalizes the NTP2 and NSOP1 variants. We explore examples and non-examples in which this lemma holds, discuss implications with syntactic properties of theories, and ask several questions.