{"paper":{"title":"The geometry and combinatorics of discrete line segment hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christopher O'Neill, Deborah Oliveros, Shira Zerbib","submitted_at":"2018-07-12T21:20:11Z","abstract_excerpt":"An $r$-segment hypergraph $H$ is a hypergraph whose edges consist of $r$ consecutive integer points on line segments in $\\mathbb{R}^2$. In this paper, we bound the chromatic number $\\chi(H)$ and covering number $\\tau(H)$ of hypergraphs in this family, uncovering several interesting geometric properties in the process. We conjecture that for $r \\ge 3$, the covering number $\\tau(H)$ is at most $(r - 1)\\nu(H)$, where $\\nu(H)$ denotes the matching number of $H$. We prove our conjecture in the case where $\\nu(H) = 1$, and provide improved (in fact, optimal) bounds on $\\tau(H)$ for $r \\le 5$. We als"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.04826","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"}