{"paper":{"title":"Many holes but no large one: maximizing $k$-holes while forbidding $(k+1)$-holes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam D\\v{z}avoronok, Aleksa D\\v{z}uklevski, Alica Dom\\'anyov\\'a, Martin Andri\\v{c}\\'ik, Matou\\v{s} \\v{S}afr\\'anek","submitted_at":"2026-06-07T18:02:36Z","abstract_excerpt":"We study the maximal number $m_{k,\\ell,n}$ of empty convex $k$-gons ($k$-holes) determined by an $n$-point set in the plane in general position that contains no empty convex $\\ell~$-gon, focusing on the first nontrivial case $\\ell=k+1$. Our main result determines the exact value in the small-excess regime: for $n=k+a$ with $a\\le k/2-1$, we prove $m_{k,k+1,k+a}=2^a.$ We also describe the extremal configurations attaining equality. Beyond this exact range, we provide upper and lower bounds in the proportional regime $n=\\alpha k$ and in the regime where $k$ is fixed and $n$ goes to infinity. In t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08762","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/2606.08762/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"}