Kaleidoscopical configurations in \(G\)-spaces (Q426765)

Summary: Let \(G\) be a group and \(X\) be a \(G\)-space with the action \(G\times X\rightarrow X, (g,x)\mapsto gx\). A subset \(F\) of \(X\) is called a kaleidoscopical configuration if there exists a coloring \(\chi:X\rightarrow C\) such that the restriction of \(\chi\) on each subset \(gF, g\in G\), is a bijection. We present a construction (called the splitting construction) of kaleidoscopical configurations in an arbitrary \(G\)-space, reduce the problem of characterization of kaleidoscopical configurations in a finite Abelian group \(G\) to a factorization of \(G\) into two subsets, and describe all kaleidoscopical configurations in isometrically homogeneous ultrametric spaces with finite distance scale. Also we construct \(2^{\mathfrak c}\) (unsplittable) kaleidoscopical configurations of cardinality \(\mathfrak c\) in the Euclidean space \(\mathbb{R}^n\).