{"paper":{"title":"On the Borsuk number of four-dimensional sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Zsolt Langi","submitted_at":"2010-07-15T09:59:58Z","abstract_excerpt":"Borsuk conjectured that every n-dimensional bounded set of positive diameter can be partitioned into n+1 sets of smaller diameters. This conjecture was proved for n=2 by Borsuk, for n=3 first by Eggleston, and disproved for n > 297 by Hinrichs and Richer. It is not known if the conjecture holds for 3 < n < 298. The best upper bound for the number of subsets of smaller diameters a four-dimensional set can be partitioned into is nine. This estimate was given by Lassak in 1982. In this note we improve this estimate by one."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2518","kind":"arxiv","version":2},"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"}