Regionally proximal relation of order dd is an equivalence one for minimal systems and a combinatorial consequence

Inverse limits of nilsystems in topologically dynamical systems were characterized by Host, Kra and Maass recently. Namely, for each d∈Nd∈N a certain generalization of the regionally proximal relation was introduced, and for a distal minimal system it was shown that such a relation is an equivalence one, which determines the maximal dd-step nilfactor. One of the main results in this article is to show that the above results hold for a general minimal system.A combinatorial consequence is also deduced, which is the topological correspondence of the result obtained by Host and Kra for positive upper Banach density subsets using ergodic methods.