Kaleidoscopical configurations in groups (Q402814)

For a set \(X\) and a family \(\mathfrak F\) of subsets of \(X\), the hypergraph \((X, \mathfrak F)\) is called kaleidoscopical if there exists a surjective coloring map such that the restriction of this map on each subset \(F\) from the family \(\mathfrak F\) is a bijection. A subset \(A\) of a group \(G\) is called a kaleidoscopical configuration if the hypergraph \((G, \{gA:g\in G\})\) is kaleidoscopical.NEWLINENEWLINEThe authors give two topological constructions of kaleidoscopical configurations. Also they prove that every infinite subset of an abelian group contains an infinite kaleidoscopical configuration.