Geodesic
Kaleidoscopical configurations in \(G\)-spaces
The electronic journal of combinatorics, Tome 19 (2012) no. 1
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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 $\mathbb{R}^n$.
DOI : 10.37236/19
Classification : 05B45, 05C15, 05E18, 20K01
Mots-clés : splitting construction of kaleidoscopical configurations 
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     author = {Taras Banakh and Oleksandr Petrenko and Igor Protasov and Sergiy Slobodianiuk},
     title = {Kaleidoscopical configurations in {\(G\)-spaces}},
     journal = {The electronic journal of combinatorics},
     year = {2012},
     volume = {19},
     number = {1},
     doi = {10.37236/19},
     zbl = {1243.05060},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/19/}
}
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Taras Banakh; Oleksandr Petrenko; Igor Protasov; Sergiy Slobodianiuk. Kaleidoscopical configurations in \(G\)-spaces. The electronic journal of combinatorics, Tome 19 (2012) no. 1. doi: 10.37236/19

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