Vapnik-Chervonenkis Dimension and Density on Johnson and Hamming Graphs

VC-dimension and VC-density are measures of combinatorial complexity of set systems. VC-dimension was first introduced in the context of statistical learning theory, and is tightly related to the sample complexity in PAC learning. VC-density is a refinement of VC-dimension. Both notions are also studied in model theory, in the context of \emph{dependent} theories. A set system that is definable by a formula of first-order logic with parameters has finite VC-dimension if and only if the formula is a dependent formula. In this paper we study the VC-dimension and the VC-density of the edge relation $Exy$ on Johnson graphs and on Hamming graphs. On a graph $G$, the set system defined by the formula $Exy$ is the vertex set of $G$ along with the collection of all \emph{open neighbourhoods} of $G$. We show that the edge relation has VC-dimension at most $4$ on Johnson graphs and at most $3$ on Hamming graphs and these bounds are optimal. We furthermore show that the VC-density of the edge relation on the class of all Johnson graphs is $2$, and on the class of all Hamming graphs the VC-density is $2$ as well. Moreover, we show that our bounds on the VC-dimension carry over to the class of all induced subgraphs of Johnson graphs, and to the class of all induced subgraphs of Hamming graphs, respectively. It also follows that the VC-dimension of the set systems of \emph{closed neighbourhoods} in Johnson graphs and Hamming graphs is bounded. Johnson graphs and Hamming graphs are well known examples of distance transitive graphs. Neither of these graph classes is nowhere dense nor is there a bound on their (local) clique-width. Our results contrast this by giving evidence of structural tameness of the graph classes.