Forking and Dividing in Henson Graphs

Abstract

For $n\ge 3$, define $T_n$ to be the theory of the generic $K_n$-free graph, where $K_n$ is the complete graph on $n$ vertices. We prove a graph-theoretic characterization of dividing in $T_n$ and use it to show that forking and dividing are the same for complete types. We then give an example of a forking and nondividing formula. Altogether, $T_n$ provides a counterexample to a question of Chernikov and Kaplan.