{"paper":{"title":"Distinguishing extension numbers for $\\mathbf R^n$ and $S^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Alex Lombardi","submitted_at":"2014-08-25T17:58:38Z","abstract_excerpt":"In the setting of a group $\\Gamma$ acting faithfully on a set $X$, a $k$-coloring $c: X\\rightarrow \\{1, 2, ..., k\\}$ is called $\\Gamma$-distinguishing if the only element of $\\Gamma$ that fixes $c$ is the identity element. The distinguishing number $D_\\Gamma(X)$ is the minimum value of $k$ such that a $\\Gamma$-distinguishing $k$-coloring of $X$ exists. Now, fixing $k= D_\\Gamma(X)$, a subset $W\\subset X$ with trivial pointwise stabilizer satisfies the precoloring extension property $P(W)$ if every precoloring $c: X-W\\rightarrow \\{1, ..., k\\}$ can be extended to a $\\Gamma$-distinguishing $k$-col"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5849","kind":"arxiv","version":1},"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"}