Model Theory of Johnson Graphs

Abstract

A \emph{Johnson graph} $J(n,k)$ is a graph whose vertices are the $k$-element subsets of an underlying set of size $n$ where two vertices are adjacent if they intersect in a subset of size $k-1$. We extend the definition to allow infinite parameters and we study a natural limit theory $T_{\mathcal{J}}$ of Johnson graphs. We show that $T_{\mathcal{J}}$ is complete and we give an axiomatisation of $T_{\mathcal{J}}$, from which we obtain a complete description of the models of $T_{\mathcal{J}}$. We use this to show tameness properties of $T_{\mathcal{J}}$, namely that $T_{\mathcal{J}}$ is $ω$-stable of Morley rank $ω$, but not monadically dependent. In particular it follows that the class of all finite Johnson graphs is stable, and therefore dependent but not monadically dependent.