{"paper":{"title":"Lines in metric spaces: universal lines counted with multiplicity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jos\\'e Zamora, Mart\\'in Matamala","submitted_at":"2018-03-19T20:36:52Z","abstract_excerpt":"The line generated by two distinct points, $x$ and $y$, in a finite metric space $M=(V,d)$, denoted by $\\overline{xy}^M$, is the set of points given by $$\\overline{xy}^M:=\\{z\\in V: d(x,y)=|d(x,z)\\pm d(z,y)|\\}.$$ A 2-set $\\{x,y\\}$ such that $\\overline{xy}^M=V$ is called a universal pair and its associated line a universal line.\n  Chen and Chv\\'atal conjectured that in any finite metric space either there is a universal line or there are at least $|V|$ different (non-universal) lines. Chv\\'atal proved that this is indeed the case when the metric space has distances in the set $\\{0,1,2\\}$.\n  Abou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.07154","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"}