A New Kim’s Lemma

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 NTP 2 and NSOP 1 theories.We introduce a new variant of Kim's Lemma that simultaneously generalizes the NTP 2 and NSOP 1 variants.We explore examples and nonexamples in which this lemma holds, discuss implications with syntactic properties of theories, and ask several questions. 1. Introduction 825 2. Preliminaries 828 3. A diversity of Kim's lemmas 838 4. Examples 844 5.