# Kaleidoscopical Configurations in G-spaces - Mathematics > Combinatorics

Abstract: Let $G$ be a group and $X$ be a $G$-space. A subset $F$ of $X$ is called akaleidoscopical configuration if there exists a surjective coloring $\chi:X\toY$ such that the restriction of $\chi$ on each subset $gF$, $g\in G$ is abijection. We give some constructions of kaleidoscopical configurations in anarbitrary $G$-space, develop some kaleidoscopical technique for Abelian groupsconsidered as $G$-spaces with the action $g,x\mapsto g+x$, and describekaleidoscopical configurations in the cyclic groups of order $N=p^m$ or$N=p 1 . p k$ where $p$ is prime and $p 1, .,p k$ are distinct primes.Let $G$ be a group and $X$ be a $G$-space. A subset $F$ of $X$ is called akaleidoscopical configuration if there exists a coloring $\chi:X ightarrow C$such that the restriction of $\chi$ on each subset $gF$, $g\in G$, is abijection. We present a construction called the splitting construction ofkaleidoscopical configurations in an arbitrary $G$-space, reduce the problem ofcharacterization of kaleidoscopical configurations in a finite Abelian group$G$ to a factorization of $G$ into two subsets, and describe allkaleidoscopical configurations in isometrically homogeneous ultrametric spaceswith finite distance scale. Also we construct $2^c$ unsplittablekaleidoscopical configurations of cardinality continuum in the Euclidean space$R^n$.

Author: T.O. Banakh, O. Petrenko, I.V. Protasov, S. Slobodianiuk

Source: https://arxiv.org/