A note on generic $n$-partite graphs

Abstract

An $n$-partite graph is a graph such that every vertex has a color in $\{1,\ldots,n\}$ and every two vertices of the same color are not adjacent. We study the model comparisons of the theories of $n$-partite graph and $K_{\overline{m}}$-free $n$-partite graph, where $K_{\overline{m}}$ is a complete graph of a given size. The model companion of the theory of $n$-partite graph is simple and has IP. The model companion of the theory of $K_{\overline{m}}$-free $n$-partite graph has $\operatorname{TP}_2$, $\operatorname{SOP}_3$ and $\operatorname{NSOP}_4$ if $n > 2$. Forking independence coincides with dividing independence in this theory.