Subgraph densities in a surface

Given a fixed graph H that embeds in a surface Σ , what is the maximum number of copies of H in an n -vertex graph G that embeds in Σ ? We show that the answer is Θ(nf(H)) , where f ( H ) is a graph invariant called the ‘flap-number’ of H , which is independent of Σ . This simultaneously answers two open problems posed by Eppstein ((1993) J. Graph Theory 17 (3) 409–416.). The same proof also answers the question for minor-closed classes. That is, if H is a K3,t minor-free graph, then the maximum number of copies of H in an n -vertex K3,t minor-free graph G is Θ(nf'(H)) , where f ′( H ) is a graph invariant closely related to the flap-number of H . Finally, when H is a complete graph we give more precise answers.