{"paper":{"title":"On $(\\le k)$-edges, crossings, and halving lines of geometric drawings of $K_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DM"],"primary_cat":"math.CO","authors_text":"Bernardo M. \\'Abrego, Gelasio Salazar, Jes\\'us Lea\\~nos, Mario Cetina, Silvia Fern\\'andez-Merchant","submitted_at":"2011-02-24T19:02:55Z","abstract_excerpt":"Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\\crg(P)$, is the \\emph{rectilinear crossing number} of $P$. A \\emph{halving line} of $P$ is a line passing though two points of $P$ that divides the rest of the points of $P$ in (almost) half. The number of halving lines of $P$ is denoted by $h(P)$. Similarly, a $k$\\emph{-edge}, $0\\leq k\\leq n/2-1$, is a line passing through two points of $P$ and leaving exactly $k$ points of $P$ on one side. The number of $(\\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.5065","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"}