Interpreting nowhere dense graph classes as a classical notion of model theory

A class of graphs is nowhere dense if for every integer r there is a finite upper bound on the size of complete graphs that occur as r -minors. We observe that this recent tameness notion from (algorithmic) graph theory is essentially the earlier stability theoretic notion of superflatness. For subgraph-closed classes of graphs we prove equivalence to stability and to not having the independence property. Expressed in terms of PAC learning, the concept classes definable in first-order logic in a subgraph-closed graph class have bounded sample complexity, if and only if the class is nowhere dense.