{"paper":{"title":"Kaleidoscopical Configurations in G-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"I.V. Protasov, O. Petrenko, S. Slobodianiuk, T.O. Banakh","submitted_at":"2010-01-06T14:14:40Z","abstract_excerpt":"Let $G$ be a group and $X$ be a $G$-space. A subset $F$ of $X$ is called a kaleidoscopical configuration if there exists a surjective coloring $\\chi:X\\to Y$ such that the restriction of $\\chi$ on each subset $gF$, $g\\in G$ is a bijection. We give some constructions of kaleidoscopical configurations in an arbitrary $G$-space, develop some kaleidoscopical technique for Abelian groups (considered as $G$-spaces with the action $(g,x)\\mapsto g+x$), and describe kaleidoscopical 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 pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.0903","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}