{"id":"https://openalex.org/W2058787718","doi":"https://doi.org/10.1137/130950422","title":"Empty Pentagons in Point Sets with Collinearities","display_name":"Empty Pentagons in Point Sets with Collinearities","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2058787718","doi":"https://doi.org/10.1137/130950422","mag":"2058787718"},"language":"en","primary_location":{"id":"doi:10.1137/130950422","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130950422","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1207.3633","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"J\u00e1nos Bar\u00e1t","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"J\u00e1nos Bar\u00e1t","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Vida Dujmovi\u0107","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Vida Dujmovi\u0107","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Gwena\u00ebl Joret","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gwena\u00ebl Joret","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Michael S. Payne","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Michael S. Payne","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Ludmila Scharf","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ludmila Scharf","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Daria Schymura","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Daria Schymura","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Pavel Valtr","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pavel Valtr","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"David R. Wood","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"David R. Wood","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":8,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.08568363,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"29","issue":"1","first_page":"198","last_page":"209"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.8260999917984009,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.8260999917984009,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.03970000147819519,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.029500000178813934,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/pentagon","display_name":"Pentagon","score":0.9659000039100647},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5906999707221985},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.5516999959945679},{"id":"https://openalex.org/keywords/general-position","display_name":"General position","score":0.5422000288963318},{"id":"https://openalex.org/keywords/convex-polytope","display_name":"Convex polytope","score":0.48019999265670776},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.45210000872612},{"id":"https://openalex.org/keywords/finite-set","display_name":"Finite set","score":0.4359000027179718},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.43540000915527344},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4307999908924103}],"concepts":[{"id":"https://openalex.org/C2779965921","wikidata":"https://www.wikidata.org/wiki/Q127840","display_name":"Pentagon","level":2,"score":0.9659000039100647},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8513000011444092},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7937999963760376},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5906999707221985},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.5516999959945679},{"id":"https://openalex.org/C150397156","wikidata":"https://www.wikidata.org/wiki/Q591310","display_name":"General position","level":2,"score":0.5422000288963318},{"id":"https://openalex.org/C110943637","wikidata":"https://www.wikidata.org/wiki/Q4129097","display_name":"Convex polytope","level":5,"score":0.48019999265670776},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.45210000872612},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.4359000027179718},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.43540000915527344},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4307999908924103},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.42820000648498535},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.4162999987602234},{"id":"https://openalex.org/C198082294","wikidata":"https://www.wikidata.org/wiki/Q3399648","display_name":"Position (finance)","level":2,"score":0.4153999984264374},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.4059000015258789},{"id":"https://openalex.org/C2777620828","wikidata":"https://www.wikidata.org/wiki/Q7582067","display_name":"Square lattice","level":3,"score":0.39719998836517334},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.3955000042915344},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3889000117778778},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3646000027656555},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.34299999475479126},{"id":"https://openalex.org/C111110010","wikidata":"https://www.wikidata.org/wiki/Q2627315","display_name":"Convex combination","level":4,"score":0.2903999984264374},{"id":"https://openalex.org/C4107886","wikidata":"https://www.wikidata.org/wiki/Q906377","display_name":"Discrete geometry","level":2,"score":0.26179999113082886},{"id":"https://openalex.org/C64051489","wikidata":"https://www.wikidata.org/wiki/Q7104528","display_name":"Orthogonal convex hull","level":5,"score":0.26080000400543213},{"id":"https://openalex.org/C7356347","wikidata":"https://www.wikidata.org/wiki/Q1307838","display_name":"Unit square","level":2,"score":0.2524000108242035},{"id":"https://openalex.org/C179117685","wikidata":"https://www.wikidata.org/wiki/Q328998","display_name":"Complex plane","level":2,"score":0.2515999972820282}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/130950422","is_oa":false,"landing_page_url":"https://doi.org/10.1137/130950422","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1207.3633","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1207.3633","pdf_url":"https://arxiv.org/pdf/1207.3633","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:dipot.ulb.ac.be:2013/207052","is_oa":false,"landing_page_url":"http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/207052","pdf_url":null,"source":{"id":"https://openalex.org/S4306401063","display_name":"D\u00e9p\u00f4t institutionnel de l'Universit\u00e9 libre de Bruxelles (Universit\u00e9 Libre de Bruxelles)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I132053463","host_organization_name":"Universit\u00e9 Libre de Bruxelles","host_organization_lineage":["https://openalex.org/I132053463"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM journal on discrete mathematics, 29 (1","raw_type":"info:ulb-repo/semantics/openurl/article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1207.3633","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1207.3633","pdf_url":"https://arxiv.org/pdf/1207.3633","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1304196372","display_name":null,"funder_award_id":"Program","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G1329855610","display_name":null,"funder_award_id":"project","funder_id":"https://openalex.org/F4320321390","funder_display_name":"Fonds De La Recherche Scientifique - FNRS"},{"id":"https://openalex.org/G2165548363","display_name":null,"funder_award_id":"Canada","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G2436386463","display_name":null,"funder_award_id":"Scient","funder_id":"https://openalex.org/F4320320875","funder_display_name":"Deutscher Akademischer Austauschdienst"},{"id":"https://openalex.org/G4952545190","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G5453931392","display_name":null,"funder_award_id":"P2020/12/G061","funder_id":"https://openalex.org/F4320321006","funder_display_name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky"},{"id":"https://openalex.org/G5704969529","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320320875","funder_display_name":"Deutscher Akademischer Austauschdienst"},{"id":"https://openalex.org/G6052429835","display_name":null,"funder_award_id":"(DFG)","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G6894402473","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8284766523","display_name":null,"funder_award_id":"(NSERC)","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8526424774","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G856527327","display_name":null,"funder_award_id":"Postdoctoral","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320315885","display_name":"Australian Government","ror":"https://ror.org/0314h5y94"},{"id":"https://openalex.org/F4320320875","display_name":"Deutscher Akademischer Austauschdienst","ror":"https://ror.org/039djdh30"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320321006","display_name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky","ror":"https://ror.org/01pv73b02"},{"id":"https://openalex.org/F4320321390","display_name":"Fonds De La Recherche Scientifique - FNRS","ror":"https://ror.org/03q83t159"},{"id":"https://openalex.org/F4320321994","display_name":"Hungarian Scientific Research Fund","ror":"https://ror.org/00v349e63"},{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W1970934335","https://openalex.org/W1977266879","https://openalex.org/W2018024429","https://openalex.org/W2072795940","https://openalex.org/W2150037097"],"related_works":[],"abstract_inverted_index":{"An":[0],"empty":[1,49,71],"pentagon":[2,50,72],"in":[3,8,17,19,29,44],"a":[4,12,60],"point":[5,26],"set":[6,13,38],"$P$":[7,18,28],"the":[9,45,64],"plane":[10,46],"is":[11,56],"of":[14,27,39],"five":[15],"points":[16,43],"strictly":[20],"convex":[21,31],"position":[22],"with":[23],"no":[24,70,74],"other":[25],"their":[30],"hull.":[32],"We":[33],"prove":[34],"that":[35],"every":[36],"finite":[37],"at":[40],"least":[41],"$328\\ell^2$":[42],"contains":[47,69],"an":[48],"or":[51],"$\\ell$":[52,75],"collinear":[53,76],"points.":[54,77],"This":[55],"optimal":[57],"up":[58],"to":[59],"constant":[61],"factor":[62],"since":[63],"$(\\ell":[65],"-1)\\times(\\ell-1)$":[66],"square":[67],"lattice":[68],"and":[73],"The":[78],"previous":[79],"best":[80],"known":[81],"bound":[82],"was":[83],"doubly":[84],"exponential.":[85]},"counts_by_year":[],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2016-06-24T00:00:00"}