Regionally proximal relation of order \(d\) is an equivalence one for minimal systems and a combinatorial consequence (Q456782)

\textit{B. Host} et al. [Adv. Math. 224, No. 1, 103--129 (2010; Zbl 1203.37022)] characterized inverse limits of nilsystems in topological dynamical systems using a topological analog of the structure theorem for measure-preserving systems. The method involves defining an appropriate generalization of the regionally proximal relation for each \(d\in\mathbb N\) and showing that this is an equivalence relation for any minimal distal system. Here this relation is shown to be an equivalence relation for minimal systems, and a combinatorial result is deduced.