{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:04:13Z","timestamp":1760241853378,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,9,29]],"date-time":"2018-09-29T00:00:00Z","timestamp":1538179200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012325","name":"National Social Science Fund of China","doi-asserted-by":"publisher","award":["18CGL018"],"award-info":[{"award-number":["18CGL018"]}],"id":[{"id":"10.13039\/501100012325","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let P be a planar point set with no three points collinear, k points of P be a k-hole of P if the k points are the vertices of a convex polygon without points of P. This article proves 13 is the smallest integer such that any planar points set containing at least 13 points with no three points collinear, contains a 3-hole, a 4-hole and a 5-hole which are pairwise disjoint.<\/jats:p>","DOI":"10.3390\/sym10100447","type":"journal-article","created":{"date-parts":[[2018,10,2]],"date-time":"2018-10-02T08:23:50Z","timestamp":1538468630000},"page":"447","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Note on the Minimum Size of a Point Set Containing Three Nonintersecting Empty Convex Polygons"],"prefix":"10.3390","volume":"10","author":[{"given":"Qing","family":"Yang","sequence":"first","affiliation":[{"name":"Faculty of Accounting, Shanxi University of Finance and Economics, Taiyuan 030006, China"}]},{"given":"Zengtai","family":"You","sequence":"additional","affiliation":[{"name":"College of Computer Science and Engineering, Dalian Minzu University, Dalian 116600, China"}]},{"given":"Xinshang","family":"You","sequence":"additional","affiliation":[{"name":"College of Economics and Management, Shandong University of Science and Technology, Qingdao 266590, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,29]]},"reference":[{"key":"ref_1","first-page":"463","article-title":"A combinatorial problem in geometry","volume":"2","author":"Szekers","year":"1935","journal-title":"Compos. Math."},{"key":"ref_2","first-page":"53","article-title":"On some extremum problem in elementary geometry","volume":"3","author":"Szekers","year":"1960","journal-title":"Ann. Univ. Sci. Budapest"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1017\/S144618110000300X","article-title":"Computer solution to the 17-point Erd\u0151s-Szekeres problem","volume":"48","author":"Szekeres","year":"2006","journal-title":"ANZIAM J."},{"key":"ref_4","first-page":"116","article-title":"Konvexe Funfeck in ebenen Punktmengen","volume":"33","author":"Harborth","year":"1978","journal-title":"Elem. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"482","DOI":"10.4153\/CMB-1983-077-8","article-title":"Sets with no empty convex 7-gons","volume":"26","author":"Horton","year":"1983","journal-title":"Can. Math. Bull."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s00454-007-9018-x","article-title":"Empty convex hexagons in planar point sets","volume":"39","author":"Gerken","year":"2008","journal-title":"Discret. Comput. Geom."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/s00454-007-1343-6","article-title":"The empty hexagon theorem","volume":"38","year":"2007","journal-title":"Discret. Comput. Geom."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/0166-218X(94)00120-3","article-title":"On a partition into convex polygons","volume":"64","author":"Urabe","year":"1996","journal-title":"Discret. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1007\/978-3-540-89550-3_10","article-title":"A minimal planar point set with specified disjoint empty convex subsets","volume":"4535","author":"Hosono","year":"2008","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0925-7721(01)00023-2","article-title":"On the number of disjoint convex quadrilaterals for a planar point set","volume":"20","author":"Hosono","year":"2001","journal-title":"Comput. Geom. Theory Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/11589440_12","article-title":"On the minimum size of a point set containing two non-intersecting empty convex polygons","volume":"3742","author":"Hosono","year":"2005","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1007\/978-3-540-70666-3_23","article-title":"Reconfirmation of two results on disjoint empty convex polygons","volume":"4381","author":"Wu","year":"2007","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_13","first-page":"445","article-title":"On the minimum size of a point set containing a 5-hole and a disjoint 4-hole","volume":"48","author":"Bhattacharya","year":"2011","journal-title":"Stud. Sci. Math. Hung."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2401","DOI":"10.1016\/j.disc.2013.07.007","article-title":"Sandip Das, On pseudo-convex partitions of a planar point set","volume":"313","author":"Bhattacharya","year":"2013","journal-title":"Discret. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s10998-013-9078-z","article-title":"Disjoint Empty Convex Pentagons in Planar Point Sets","volume":"66","author":"Bhattacharya","year":"2013","journal-title":"Period. Math. Hung."},{"key":"ref_16","unstructured":"Hosono, K., and Urabe, M. (2018). Specifed holes with pairwise disjoint interiors in planar point sets. AKCE Int. J. Graph. Comb."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1134\/S0001434615050314","article-title":"On the minimum size of a point set containing a 5-hole and double disjoint 3-holes","volume":"97","author":"You","year":"2013","journal-title":"Math. Notes"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1134\/S0001434614070281","article-title":"A note on the upper bound for disjoint convex partitions","volume":"96","author":"You","year":"2014","journal-title":"Math. Notes"},{"key":"ref_19","first-page":"459","article-title":"A note on the value about a disjoint convex partition problem","volume":"115","author":"You","year":"2014","journal-title":"Ars Comb."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1134\/S0001434618070143","article-title":"A note on the value about a disjoint convex partition problem","volume":"104","author":"You","year":"2018","journal-title":"Math. Notes"},{"key":"ref_21","first-page":"22","article-title":"On Erd\u0151s-Szekeres problem for empty hexagons in the plane","volume":"16","author":"Koshelev","year":"2009","journal-title":"Modelirovanie i Analiz Informatsionnykh Sistem"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/10\/447\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:23:07Z","timestamp":1760196187000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/10\/447"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,29]]},"references-count":21,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2018,10]]}},"alternative-id":["sym10100447"],"URL":"https:\/\/doi.org\/10.3390\/sym10100447","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,9,29]]}}}