{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:36:47Z","timestamp":1787323007660,"version":"build-2736575974"},"reference-count":14,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>An empty pentagon in a point set $P$ in the plane is a set of five points in $P$ in strictly convex position with no other point of $P$ in their convex hull. We prove that every finite set of at least $328\\ell^2$ points in the plane contains an empty pentagon or $\\ell$ collinear points. This is optimal up to a constant factor since the $(\\ell -1)\\times(\\ell-1)$ square lattice contains no empty pentagon and no $\\ell$ collinear points. The previous best known bound was doubly exponential.<\/jats:p>","DOI":"10.1137\/130950422","type":"journal-article","created":{"date-parts":[[2015,1,29]],"date-time":"2015-01-29T09:38:10Z","timestamp":1422524290000},"page":"198-209","source":"Crossref","is-referenced-by-count":0,"title":["Empty Pentagons in Point Sets with Collinearities"],"prefix":"10.1137","volume":"29","author":[{"given":"J\u00e1nos","family":"Bar\u00e1t","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vida","family":"Dujmovi\u0107","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gwena\u00ebl","family":"Joret","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael S.","family":"Payne","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ludmila","family":"Scharf","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daria","family":"Schymura","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pavel","family":"Valtr","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David R.","family":"Wood","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,1,29]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-010-0957-2"},{"key":"atypb2","unstructured":"J. Cibulka and J. Kyncl,\n                      Private Communication\n                      , 2012."},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"J. Cibulka, J. Kyncl, and P. Valtr,\n                      On planar point sets with the pentagon property\n                      , in Proceedings of the 29th Annual Symposium on Computational Geometry, SoCG '13, New York, 2013, ACM, pp. 81-90.","DOI":"10.1145\/2493132.2462406"},{"key":"atypb4","first-page":"3","volume":"1","author":"Eppstein D.","year":"2010","journal-title":"J. Comput. Geom."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02414146"},{"key":"atypb6","first-page":"463","volume":"2","author":"Erdo\u0308s P.","year":"1935","journal-title":"Compos. Math."},{"key":"atypb7","unstructured":"P. Erdo\u0308s and G. Szekeres,\n                      On some extremum problems in elementary geometry\n                      , Ann. Univ. Sci. Budapest. Eo\u0308tvo\u0308s Sect. Math., 3-4 (1960\/1961), pp. 53-62."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-007-9018-x"},{"key":"atypb9","first-page":"116","volume":"33","author":"Harborth H.","year":"1978","journal-title":"Elem. Math."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1983-077-8"},{"key":"atypb11","first-page":"734","volume":"415","author":"Koshelev V. A.","year":"2007","journal-title":"Dokl. Akad. 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