{"paper":{"title":"Covering half-grids with lines and planes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Anurag Bishnoi, Shantanu Nene","submitted_at":"2025-01-19T20:01:04Z","abstract_excerpt":"We study hyperplane covering problems for finite grid-like structures in $\\mathbb{R}^d$. We call a set $\\mathcal{C}$ of points in $\\mathbb{R}^2$ a conical grid if the line $y = a_i$ intersects $\\mathcal{C}$ in exactly $i$ points, for some $a_1 > \\cdots > a_n \\in \\mathbb{R}$. We prove that the number of lines required to cover every point of such a grid at least $k$ times is at least $nk\\left(1-\\frac{1}{e}-O(\\frac{1}{n}) \\right)$. If the grid $\\mathcal{C}$ is obtained by cutting an $m \\times n$ grid of points in half along one of the diagonals, then we prove the lower bound of $mk\\left(1-e^{-\\f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.11156","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.11156/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}