{"id":"https://openalex.org/W7224821987","doi":"https://doi.org/10.48550/arxiv.2610.39311","title":"Tight Asymptotic Bounds on the Size of Order Graphs","display_name":"Tight Asymptotic Bounds on the Size of Order Graphs","publication_year":2026,"publication_date":"2026-10-29","ids":{"openalex":"https://openalex.org/W7224821987","doi":"https://doi.org/10.48550/arxiv.2610.39311"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.39311","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.39311","pdf_url":"https://arxiv.org/pdf/2610.39311","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.39311","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5017265124","display_name":"Ruben Becker","orcid":"https://orcid.org/0000-0002-3495-3753","observed_orcids":["https://orcid.org/0000-0002-3495-3753"]},"institutions":[{"id":"https://openalex.org/I149461666","display_name":"Ca' Foscari University of Venice","ror":"https://ror.org/04yzxz566","country_code":"IT","type":"education","lineage":["https://openalex.org/I149461666"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Becker, Ruben","raw_affiliation_strings":["Ca' Foscari University of Venice , Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ca' Foscari University of Venice , Italy","institution_ids":["https://openalex.org/I149461666"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5045121904","display_name":"Adri\u00e1n G\u00f3mez\u2010Brand\u00f3n","orcid":"https://orcid.org/0000-0002-1216-2176","observed_orcids":["https://orcid.org/0000-0002-1216-2176"]},"institutions":[{"id":"https://openalex.org/I11019714","display_name":"Universidade da Coru\u00f1a","ror":"https://ror.org/01qckj285","country_code":"ES","type":"education","lineage":["https://openalex.org/I11019714"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"G\u00f3mez-Brand\u00f3n, Adri\u00e1n","raw_affiliation_strings":["Universidade da Coru\u00f1a & IMFD , Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universidade da Coru\u00f1a & IMFD , Spain","institution_ids":["https://openalex.org/I11019714"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080743153","display_name":"Gonzalo Navarro","orcid":"https://orcid.org/0000-0002-2286-741X","observed_orcids":["https://orcid.org/0000-0002-2286-741X"]},"institutions":[{"id":"https://openalex.org/I69737025","display_name":"University of Chile","ror":"https://ror.org/047gc3g35","country_code":"CL","type":"education","lineage":["https://openalex.org/I69737025"]}],"countries":["CL"],"is_corresponding":false,"raw_author_name":"Navarro, Gonzalo","raw_affiliation_strings":["Universidad de Chile , DCC & IMFD , Chile "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universidad de Chile , DCC & IMFD , Chile ","institution_ids":["https://openalex.org/I69737025"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5007324430","display_name":"Nicola Prezza","orcid":"https://orcid.org/0000-0003-3553-4953","observed_orcids":["https://orcid.org/0000-0003-3553-4953"]},"institutions":[{"id":"https://openalex.org/I149461666","display_name":"Ca' Foscari University of Venice","ror":"https://ror.org/04yzxz566","country_code":"IT","type":"education","lineage":["https://openalex.org/I149461666"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Prezza, Nicola","raw_affiliation_strings":["Ca' Foscari University of Venice , Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ca' Foscari University of Venice , Italy","institution_ids":["https://openalex.org/I149461666"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.45579999685287476,"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"}},"topics":[{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.45579999685287476,"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"}},{"id":"https://openalex.org/T10317","display_name":"Advanced Database Systems and Queries","score":0.36959999799728394,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.05570000037550926,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/join-queries","display_name":"join queries","score":0.5849999785423279},{"id":"https://openalex.org/keywords/asymptotic-estimates","display_name":"asymptotic bounds","score":0.7039999961853027},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.7450000047683716},{"id":"https://openalex.org/keywords/boolean-lattice","display_name":"Boolean lattice","score":1.0}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8048999905586243},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.75},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6690999865531921},{"id":"https://openalex.org/C110546421","wikidata":"https://www.wikidata.org/wiki/Q1315869","display_name":"Arity","level":2,"score":0.582099974155426},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5350000262260437},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.46709999442100525},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.4499000012874603},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4323999881744385},{"id":"https://openalex.org/C122818955","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independent set","level":3,"score":0.391400009393692},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.3630000054836273},{"id":"https://openalex.org/C138890893","wikidata":"https://www.wikidata.org/wiki/Q620006","display_name":"Order type","level":2,"score":0.3625999987125397},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.3492000102996826},{"id":"https://openalex.org/C2778692605","wikidata":"https://www.wikidata.org/wiki/Q4041866","display_name":"Joins","level":2,"score":0.3488999903202057},{"id":"https://openalex.org/C159254197","wikidata":"https://www.wikidata.org/wiki/Q1144915","display_name":"Lexicographical order","level":2,"score":0.3481000065803528},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.3400000035762787},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.3273000121116638},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.31470000743865967},{"id":"https://openalex.org/C58315980","wikidata":"https://www.wikidata.org/wiki/Q179436","display_name":"Asymptote","level":2,"score":0.3001999855041504},{"id":"https://openalex.org/C132569581","wikidata":"https://www.wikidata.org/wiki/Q5067368","display_name":"Treewidth","level":5,"score":0.30000001192092896},{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.29600000381469727},{"id":"https://openalex.org/C65236422","wikidata":"https://www.wikidata.org/wiki/Q173740","display_name":"Cartesian product","level":2,"score":0.27900001406669617},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.27619999647140503},{"id":"https://openalex.org/C198414033","wikidata":"https://www.wikidata.org/wiki/Q5155607","display_name":"Comparability graph","level":5,"score":0.2728999853134155},{"id":"https://openalex.org/C117160843","wikidata":"https://www.wikidata.org/wiki/Q338652","display_name":"Sublinear function","level":2,"score":0.26969999074935913},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.26010000705718994},{"id":"https://openalex.org/C2984853995","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independence number","level":3,"score":0.2590999901294708}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.39311","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.39311","pdf_url":"https://arxiv.org/pdf/2610.39311","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"doi:10.48550/arxiv.2610.39311","doi":"https://doi.org/10.48550/arxiv.2610.39311","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.39311","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.39311","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.39311","pdf_url":"https://arxiv.org/pdf/2610.39311","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7224821987.pdf","grobid_xml":"https://content.openalex.org/works/W7224821987.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2396405335","https://openalex.org/W7104033818","https://openalex.org/W2047141437","https://openalex.org/W2400630495","https://openalex.org/W7163915602","https://openalex.org/W2603475120","https://openalex.org/W2885154942","https://openalex.org/W143078694","https://openalex.org/W6903752663","https://openalex.org/W2003860402"],"abstract_inverted_index":{"Order":[0],"graphs":[1,112],"were":[2],"introduced":[3],"by":[4],"Arroyuelo":[5],"et":[6],"al.\\":[7],"(TODS":[8],"2024)":[9],"as":[10,29,32],"a":[11,124,195],"generalization":[12],"of":[13,24,49,51,62,100,108,120,180,198],"the":[14,34,47,59,72,98,117,159,162,181,199],"ring":[15],"index,":[16],"to":[17,46,78],"support":[18],"worst-case":[19],"optimal":[20],"joins":[21],"on":[22,176],"relations":[23],"arity":[25],"$d$":[26],"while":[27],"storing":[28],"few":[30],"columns":[31],"possible:":[33],"index":[35],"stores":[36],"one":[37],"column":[38],"per":[39],"edge,":[40],"so":[41,150],"its":[42],"size":[43,164],"is":[44,146,154,165,169,193],"proportional":[45],"number":[48,61],"edges":[50,63,92],"an":[52,86,172],"order":[53,87,111,133],"graph.":[54],"It":[55],"was":[56],"known":[57],"that":[58,97,101,129],"minimum":[60,163],"lies":[64],"between":[65],"$\u03a9(2^d\\sqrt":[66],"d)$":[67],"and":[68,71,95,127,202,205],"$O(2^d":[69],"d)$,":[70],"lower":[73,102,200],"bound":[74,103,145,192],"had":[75],"been":[76],"claimed":[77],"be":[79,114,137],"$42$":[80],"for":[81,93,106,131,148],"$d=6$.":[82],"We":[83,122,156],"first":[84],"exhibit":[85],"graph":[88,153],"with":[89,184],"only":[90,105],"$40$":[91,147],"$d=6$,":[94,149],"show":[96],"proof":[99,128],"holds":[104,130],"unions":[107],"cycles,":[109],"whereas":[110],"can":[113],"smaller":[115],"than":[116,208],"smallest":[118],"set":[119],"cycles.":[121],"give":[123],"corrected":[125],"statement":[126],"every":[132],"graph:":[134],"there":[135],"must":[136],"at":[138,185],"least":[139],"$\\max_m\\binom":[140],"dm\\lceil":[141],"m/2\\rceil$":[142],"edges.":[143,189],"This":[144,168],"our":[151],"$40$-edge":[152],"optimal.":[155],"then":[157],"settle":[158],"asymptotic":[160],"question:":[161],"$\u0398(2^d\\sqrt":[166],"d)$.":[167],"obtained":[170],"via":[171],"explicit":[173],"construction,":[174],"based":[175],"symmetric":[177],"chain":[178],"decompositions":[179],"Boolean":[182],"lattice,":[183],"most":[186],"$(d-1)\\binom{d}{\\lfloor":[187],"d/2\\rfloor}+2^d$":[188],"The":[190],"upper":[191],"within":[194],"factor":[196],"$4+O(d^{-1/2})$":[197],"bound,":[201],"both":[203],"numerically":[204],"asymptotically":[206],"better":[207],"trie":[209],"switching.":[210]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-31T00:00:00"}