{"paper":{"title":"Randomised algebraic constructions for the no-$(k+1)$-in-line problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benedek Kov\\'acs, D\\'avid R. Szab\\'o, Zolt\\'an L\\'or\\'ant Nagy","submitted_at":"2025-08-11T05:25:37Z","abstract_excerpt":"The no-(k+1)-in line problem seeks the maximum number of points that can be selected from an $n \\times n$ square lattice such that no $k+1$ of them are collinear. The problem was first posed more than $100$ years ago for the special case $k=2$ and has remained open ever since. The general problem was recently resolved in the case $k$ is not small compared to $n$, as Kov\\'acs, Nagy and Szab\\'o proved that the upper bound $kn$ can be attained, provided that $k>C\\sqrt{n\\log{n}}$ for an absolute constant $C$.\n  In this paper, we show that $\\left(1-\\tfrac{2}{k}\\right)kn \\leq f_k(n)\\leq kn$ and $\\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.07632","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/2508.07632/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"}