{"id":"https://openalex.org/W1974558861","doi":"https://doi.org/10.1137/090767947","title":"Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons","display_name":"Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W1974558861","doi":"https://doi.org/10.1137/090767947","mag":"1974558861"},"language":"en","primary_location":{"id":"doi:10.1137/090767947","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090767947","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":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5091823049","display_name":"Oswin Aichholzer","orcid":"https://orcid.org/0000-0002-2364-0583"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Oswin Aichholzer","raw_affiliation_strings":["oaich@ist.tugraz.at and thackl@ist.tugraz.at and bvogt@ist.tugraz.at#TAB#"],"affiliations":[{"raw_affiliation_string":"oaich@ist.tugraz.at and thackl@ist.tugraz.at and bvogt@ist.tugraz.at#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103249365","display_name":"Thomas Hackl","orcid":"https://orcid.org/0000-0002-9875-9415"},"institutions":[{"id":"https://openalex.org/I4092182","display_name":"Graz University of Technology","ror":"https://ror.org/00d7xrm67","country_code":"AT","type":"education","lineage":["https://openalex.org/I4092182"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Thomas Hackl","raw_affiliation_strings":[", Graz University of Technology#TAB#"],"affiliations":[{"raw_affiliation_string":", Graz University of Technology#TAB#","institution_ids":["https://openalex.org/I4092182"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032873622","display_name":"Clemens Huemer","orcid":"https://orcid.org/0000-0001-7557-0823"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Clemens Huemer","raw_affiliation_strings":["clemens.huemer@upc.edu#TAB#"],"affiliations":[{"raw_affiliation_string":"clemens.huemer@upc.edu#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111463493","display_name":"Ferr\u00e1n Hurtado","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ferran Hurtado","raw_affiliation_strings":["ferran.hurtado@upc.edu#TAB#"],"affiliations":[{"raw_affiliation_string":"ferran.hurtado@upc.edu#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073288056","display_name":"Birgit Vogtenhuber","orcid":"https://orcid.org/0000-0002-7166-4467"},"institutions":[{"id":"https://openalex.org/I4092182","display_name":"Graz University of Technology","ror":"https://ror.org/00d7xrm67","country_code":"AT","type":"education","lineage":["https://openalex.org/I4092182"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Birgit Vogtenhuber","raw_affiliation_strings":[", Graz University of Technology#TAB#"],"affiliations":[{"raw_affiliation_string":", Graz University of Technology#TAB#","institution_ids":["https://openalex.org/I4092182"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5091823049"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.3455,"has_fulltext":false,"cited_by_count":21,"citation_normalized_percentile":{"value":0.93051275,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"23","issue":"4","first_page":"2147","last_page":"2155"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9976999759674072,"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":0.9976999759674072,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9854999780654907,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9670000076293945,"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/monochromatic-color","display_name":"Monochromatic color","score":0.8764048218727112},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8545793294906616},{"id":"https://openalex.org/keywords/general-position","display_name":"General position","score":0.8299387693405151},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7805827856063843},{"id":"https://openalex.org/keywords/quadrilateral","display_name":"Quadrilateral","score":0.7722171545028687},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5874747037887573},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.5427063703536987},{"id":"https://openalex.org/keywords/position","display_name":"Position (finance)","score":0.528016984462738},{"id":"https://openalex.org/keywords/polygon","display_name":"Polygon (computer graphics)","score":0.5106773972511292},{"id":"https://openalex.org/keywords/colored","display_name":"Colored","score":0.5009822845458984},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.4507173001766205},{"id":"https://openalex.org/keywords/convex-polygon","display_name":"Convex polygon","score":0.4304388761520386},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1253601610660553},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12034738063812256},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.05670362710952759},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.054481685161590576}],"concepts":[{"id":"https://openalex.org/C40833965","wikidata":"https://www.wikidata.org/wiki/Q6901438","display_name":"Monochromatic color","level":2,"score":0.8764048218727112},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8545793294906616},{"id":"https://openalex.org/C150397156","wikidata":"https://www.wikidata.org/wiki/Q591310","display_name":"General position","level":2,"score":0.8299387693405151},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7805827856063843},{"id":"https://openalex.org/C151756577","wikidata":"https://www.wikidata.org/wiki/Q36810","display_name":"Quadrilateral","level":3,"score":0.7722171545028687},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5874747037887573},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.5427063703536987},{"id":"https://openalex.org/C198082294","wikidata":"https://www.wikidata.org/wiki/Q3399648","display_name":"Position (finance)","level":2,"score":0.528016984462738},{"id":"https://openalex.org/C190694206","wikidata":"https://www.wikidata.org/wiki/Q3276654","display_name":"Polygon (computer graphics)","level":3,"score":0.5106773972511292},{"id":"https://openalex.org/C2778307483","wikidata":"https://www.wikidata.org/wiki/Q5149038","display_name":"Colored","level":2,"score":0.5009822845458984},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.4507173001766205},{"id":"https://openalex.org/C2780034980","wikidata":"https://www.wikidata.org/wiki/Q3259505","display_name":"Convex polygon","level":3,"score":0.4304388761520386},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1253601610660553},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12034738063812256},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.05670362710952759},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.054481685161590576},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0},{"id":"https://openalex.org/C126042441","wikidata":"https://www.wikidata.org/wiki/Q1324888","display_name":"Frame (networking)","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/090767947","is_oa":false,"landing_page_url":"https://doi.org/10.1137/090767947","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"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.6399999856948853,"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W134663479","https://openalex.org/W173999961","https://openalex.org/W1487872891","https://openalex.org/W1539895965","https://openalex.org/W1890590972","https://openalex.org/W1967235411","https://openalex.org/W1977266879","https://openalex.org/W1993401990","https://openalex.org/W2015735942","https://openalex.org/W2018024429","https://openalex.org/W2071873201","https://openalex.org/W2072795940","https://openalex.org/W2083084150","https://openalex.org/W2108355214","https://openalex.org/W2134189835","https://openalex.org/W2164902489","https://openalex.org/W2318992515","https://openalex.org/W2404013119","https://openalex.org/W2507020368","https://openalex.org/W3159335750","https://openalex.org/W4240657212"],"related_works":["https://openalex.org/W2108355214","https://openalex.org/W4221140286","https://openalex.org/W2962748612","https://openalex.org/W2047412641","https://openalex.org/W1974558861","https://openalex.org/W2090067893","https://openalex.org/W2048256194","https://openalex.org/W2883608652","https://openalex.org/W3159335750","https://openalex.org/W2060633701"],"abstract_inverted_index":{"We":[0,98],"consider":[1],"a":[2,5,18,38,48,64,81],"variation":[3],"of":[4,17,26,40,47,56,93,106],"problem":[6],"stated":[7],"by":[8],"Erd\u0151s":[9],"and":[10,60],"Szekeres":[11],"in":[12,31,34,95,108,110],"1935":[13],"about":[14],"the":[15,35,45,54,77],"existence":[16],"number":[19],"$f^{\\mathrm{ES}}(k)$":[20,29],"such":[21],"that":[22,43,63,100],"any":[23,91,101],"set":[24,105],"S":[25,57,94],"at":[27,114],"least":[28,115],"points":[30,42,55,92,107],"general":[32,111],"position":[33,112],"plane":[36],"has":[37],"subset":[39],"k":[41],"are":[44,58],"vertices":[46,75],"convex":[49],"k-gon.":[50],"In":[51],"our":[52],"setting":[53],"colored,":[59],"we":[61],"say":[62],"(not":[65],"necessarily":[66],"convex)":[67],"spanned":[68],"polygon":[69,82],"is":[70,83],"monochromatic":[71,118],"if":[72,86],"all":[73],"its":[74,96],"have":[76],"same":[78],"color.":[79],"Moreover,":[80],"called":[84],"empty":[85],"it":[87],"does":[88],"not":[89],"contain":[90],"interior.":[97],"show":[99],"sufficiently":[102],"large":[103],"bichromatic":[104],"$\\mathbb{R}^2$":[109],"determines":[113],"one":[116],"empty,":[117],"quadrilateral":[119],"(and":[120],"thus":[121],"linearly":[122],"many).":[123]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
