{"paper":{"title":"Splitting multidimensional necklaces and measurable colorings of Euclidean spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jaros{\\l}aw Grytczuk, Wojciech Lubawski","submitted_at":"2012-09-09T16:23:27Z","abstract_excerpt":"A necklace splitting theorem of Goldberg and West asserts that any k-colored (continuous) necklace can be fairly split using at most k cuts. Motivated by the problem of Erd\\H{o}s on strongly nonrepetitive sequences, Alon et al. proved that there is a (t+3)-coloring of the real line in which no necklace has a fair splitting using at most t cuts. We generalize this result for higher dimensional spaces. More specifically, we prove that there is k-coloring of R^{d} such that no cube has a fair splitting of size t (using at most t hyperplanes orthogonal to each of the axes), provided k>(t+4)^{d}-(t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1809","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"}