{"paper":{"title":"Towards a de Bruijn-Erd\\H os theorem in the $L_1$-metric","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Balazs Patkos, Ida Kantor","submitted_at":"2012-07-16T14:16:07Z","abstract_excerpt":"A well-known theorem of de Bruijn and Erd\\H{o}s states that any set of $n$ non-collinear points in the plane determines at least $n$ lines. Chen and Chv\\'{a}tal asked whether an analogous statement holds within the framework of finite metric spaces, with lines defined using the notion of {\\em betweenness}.\n  In this paper, we prove that the answer is affirmative for sets of $n$ points in the plane with the $L_1$ metric, provided that no two points share their $x$- or $y$-coordinate. In this case, either there is a line that contains all $n$ points, or $X$ induces at least $n$ distinct lines.\n "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3688","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"}