Local character of Kim-independence

Abstract

We show that NSOP$_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is NSOP$_{1}$, $M\models T$, and $p$ is a type over $M$, then the collection of elementary substructures of size $\left|T\right|$ over which $p$ does not Kim-fork is a club of $\left[M\right]^{\left|T\right|}$ and that this characterizes NSOP$_{1}$. We also present a new phenomenon we call dual local-character for Kim-independence in NSOP$_{1}$-theories.