{"paper":{"title":"Blow-ups of order types of positive density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Benedikt Hahn, Jes\\'us Lea\\~nos, Ruy Fabila-Monroy","submitted_at":"2026-06-05T19:34:47Z","abstract_excerpt":"Order types are an equivalence relation between point configurations that capture their combinatorial and convexity properties. Let $P$ be a $\\kappa$-colored sequence of $n \\ge d+1$ points in general position in $\\mathbb{R}^d$. Let $\\rho$ be a $\\kappa$-colored order type on $k \\le d+1$ points that has positive density on $P$; that is, for some constant $\\delta >0$, there are $\\delta \\cdot \\binom{n}{k}$ $k$-point subsequences of $P$ that have the same order type as $\\rho$ and the same color pattern. In this paper we show that there exists a constant $c >0$ (depending only on $d, \\delta$, $k$ an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07806","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/2606.07806/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"}