{"paper":{"title":"The extensible no-$(k(n)+1)$-in-line problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benedek N\\'ador, D\\'avid Melj\\'an, M\\'at\\'e J\\'anosik, Tam\\'as G\\'abriel","submitted_at":"2026-06-01T20:07:01Z","abstract_excerpt":"The classical no-$k$-in-line problem asks for the largest number of points that can be placed on an $n \\times n$ grid without having $k$ of them collinear. A natural extension, motivated by the analogous question by Erde for $k\\in \\mathbb{Z}$, is the \\emph{extensible no-$(k(n)+1)$-in-line problem}, which seeks a subset of points in $\\mathbb{Z}^2$ with maximal possible density such that at most $k(n)$ points are collinear within the subgrid $[1,n]^2$.\n  We construct optimal sets for linear functions and positive-density sets for power functions. We prove that any configuration achieving $\\limin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02843","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/2606.02843/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"}