Perfect independent sets with respect to infinitely many relations (Q334980)

Let \(\kappa\) be an infinite cardinal number and let \(X\) be a completely metrizable space of weight~\(\kappa\). Furthermore let \(\mathcal{R}\) be a countable family of relations on~\(X\), each of some arity and an \(F_\sigma\)-subset of the corresponding power. Then, one of the following holds: there exists a perfect set that is independent with respect to all relations in~\(\mathcal{R}\), or there is an ordinal~\(\gamma\) below~\(\kappa^+\) such that every such independent set has Cantor-Bendixson rank not more than~\(\gamma\). The authors actually prove the dual form on cliques with respect to \(G_\delta\)-relations. There are applications of this result to topological groups: sufficient conditions for the existence of various subgroups with perfect sets of generators.