{"paper":{"title":"On the Existence of Ordinary Triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hossein Nassajian Mojarrad, J\\'ozsef Solymosi, M\\'arton Nasz\\'odi, May Szedl\\'ak, Radoslav Fulek, Sebastian U. Stich","submitted_at":"2017-01-27T20:02:04Z","abstract_excerpt":"Let $P$ be a finite point set in the plane. A \\emph{$c$-ordinary triangle} in $P$ is a subset of $P$ consisting of three non-collinear points such that each of the three lines determined by the three points contains at most $c$ points of $P$. Motivated by a question of Erd\\H{o}s, and answering a question of de Zeeuw, we prove that there exists a constant $c>0$ such that $P$ contains a $c$-ordinary triangle, provided that $P$ is not contained in the union of two lines. Furthermore, the number of $c$-ordinary triangles in $P$ is $\\Omega(|P|)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08183","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}