{"paper":{"title":"Covering triangular grids with multiplicity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abdul Basit, Alexander Clifton, Paul Horn","submitted_at":"2023-07-25T04:59:34Z","abstract_excerpt":"Motivated by classical work of Alon and F\\\"uredi, we introduce and address the following problem: determine the minimum number of affine hyperplanes in $\\mathbb{R}^d$ needed to cover every point of the triangular grid $T_d(n) := \\{(x_1,\\dots,x_d)\\in\\mathbb{Z}_{\\ge 0}^d\\mid x_1+\\dots+x_d\\le n-1\\}$ at least $k$ times. For $d = 2$, we solve the problem exactly for $k \\leq 4$, and obtain a partial solution for $k > 4$. We also obtain an asymptotic formula (in $n$) for all $d \\geq k - 2$. The proofs rely on combinatorial arguments and linear programming."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.13257","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.13257/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"}