{"id":"https://openalex.org/W2163226450","doi":"https://doi.org/10.1137/140954519","title":"Distinct Volume Subsets","display_name":"Distinct Volume Subsets","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2163226450","doi":"https://doi.org/10.1137/140954519","mag":"2163226450"},"language":"en","primary_location":{"id":"doi:10.1137/140954519","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140954519","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":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1401.6734","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5042722711","display_name":"David Conlon","orcid":"https://orcid.org/0000-0001-5899-1829"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"David Conlon","raw_affiliation_strings":["University of Oxford"],"affiliations":[{"raw_affiliation_string":"University of Oxford","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5112361944","display_name":"Jacob Fox","orcid":"https://orcid.org/0000-0002-0664-497X"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jacob Fox","raw_affiliation_strings":["Massachusetts Institute Of Technology#TAB#"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute Of Technology#TAB#","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089274175","display_name":"William Gasarch","orcid":"https://orcid.org/0000-0003-3698-5991"},"institutions":[{"id":"https://openalex.org/I66946132","display_name":"University of Maryland, College Park","ror":"https://ror.org/047s2c258","country_code":"US","type":"education","lineage":["https://openalex.org/I66946132"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"William Gasarch","raw_affiliation_strings":["University of Maryland - College Park"],"affiliations":[{"raw_affiliation_string":"University of Maryland - College Park","institution_ids":["https://openalex.org/I66946132"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101783327","display_name":"David G. Harris","orcid":"https://orcid.org/0000-0002-3021-3555"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"David G. Harris","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083331079","display_name":"Douglas Ulrich","orcid":"https://orcid.org/0000-0001-5843-5400"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Douglas Ulrich","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5041338991","display_name":"Samuel Zbarsky","orcid":"https://orcid.org/0000-0001-5820-2332"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Samuel Zbarsky","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5042722711"],"corresponding_institution_ids":["https://openalex.org/I40120149"],"apc_list":null,"apc_paid":null,"fwci":0.6774,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.67927383,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"29","issue":"1","first_page":"472","last_page":"480"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9991000294685364,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9991000294685364,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9983000159263611,"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":0.9936000108718872,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8343513011932373},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.7384099960327148},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6420468091964722},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.6247278451919556},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5165585279464722},{"id":"https://openalex.org/keywords/volume","display_name":"Volume (thermodynamics)","score":0.49916958808898926},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.47979408502578735},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.4347216486930847},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3491917550563812},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2340204119682312},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09683313965797424},{"id":"https://openalex.org/keywords/biology","display_name":"Biology","score":0.046622276306152344}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8343513011932373},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.7384099960327148},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6420468091964722},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.6247278451919556},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5165585279464722},{"id":"https://openalex.org/C20556612","wikidata":"https://www.wikidata.org/wiki/Q4469374","display_name":"Volume (thermodynamics)","level":2,"score":0.49916958808898926},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.47979408502578735},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.4347216486930847},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3491917550563812},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2340204119682312},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09683313965797424},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.046622276306152344},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":9,"locations":[{"id":"doi:10.1137/140954519","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140954519","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:1401.6734","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1401.6734","pdf_url":"https://arxiv.org/pdf/1401.6734","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":"","raw_type":null},{"id":"pmh:oai:authors.library.caltech.edu:97815","is_oa":true,"landing_page_url":"https://authors.library.caltech.edu/97815/","pdf_url":"https://resolver.caltech.edu/CaltechAUTHORS:20190812-162958151","source":{"id":"https://openalex.org/S4306402161","display_name":"CaltechAUTHORS (California Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122411786","host_organization_name":"California Institute of Technology","host_organization_lineage":["https://openalex.org/I122411786"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"mag:2163226450","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1401.6734.pdf","pdf_url":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.712.6708","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.712.6708","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://people.maths.ox.ac.uk/%7Econlond/distinctvol.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.754.5989","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.754.5989","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1401.6734.pdf","raw_type":"text"},{"id":"pmh:oai:dspace.mit.edu:1721.1/97231","is_oa":true,"landing_page_url":"http://hdl.handle.net/1721.1/97231","pdf_url":null,"source":{"id":"https://openalex.org/S4306400425","display_name":"DSpace@MIT (Massachusetts Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I63966007","host_organization_name":"Massachusetts Institute of Technology","host_organization_lineage":["https://openalex.org/I63966007"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Society for Industrial and Applied Mathematics","raw_type":"http://purl.org/eprint/type/JournalArticle"},{"id":"doi:10.48550/arxiv.1401.6734","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1401.6734","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"},{"id":"mag:2236355028","is_oa":false,"landing_page_url":"http://dspace.mit.edu/handle/1721.1/97231","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1401.6734","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1401.6734","pdf_url":"https://arxiv.org/pdf/1401.6734","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":"","raw_type":null},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2498162777","display_name":"Problems in Ramsey theory and extremal combinatorics","funder_award_id":"1069197","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W1551919361","https://openalex.org/W1979578534","https://openalex.org/W1981515128","https://openalex.org/W1988090433","https://openalex.org/W2006073641","https://openalex.org/W2031397195","https://openalex.org/W2071116836","https://openalex.org/W2072637582","https://openalex.org/W2083644514","https://openalex.org/W2085391599","https://openalex.org/W2094422067","https://openalex.org/W2098884097","https://openalex.org/W2101788557","https://openalex.org/W2121195626","https://openalex.org/W2319149236","https://openalex.org/W2949342774","https://openalex.org/W2963234396","https://openalex.org/W4211205179"],"related_works":["https://openalex.org/W2236355028","https://openalex.org/W3098671307","https://openalex.org/W1996363709","https://openalex.org/W2155187680","https://openalex.org/W2949585991","https://openalex.org/W2043060962","https://openalex.org/W3038113379","https://openalex.org/W2024239729","https://openalex.org/W3002884261","https://openalex.org/W3009078167","https://openalex.org/W1918058354","https://openalex.org/W2963311461","https://openalex.org/W1979382580","https://openalex.org/W1517049013","https://openalex.org/W1534828312","https://openalex.org/W1976681187","https://openalex.org/W3123525099","https://openalex.org/W2050820552","https://openalex.org/W2952870569","https://openalex.org/W2019903106"],"abstract_inverted_index":{"Suppose":[0],"that":[1,20,84],"$a$":[2,48,95],"and":[3,63,82,96],"$d$":[4],"are":[5,49],"positive":[6],"integers":[7],"with":[8,52],"$a":[9],"\\geq":[10],"2$.":[11],"Let":[12],"$h_{a,d}(n)$":[13,85],"be":[14,67],"the":[15,37,41,56,76],"largest":[16],"integer":[17],"$t$":[18,32],"such":[19],"any":[21],"set":[22],"of":[23,31,40,46,72,91],"$n$":[24,92],"points":[25,33],"in":[26,54],"$\\mathbb{R}^d$":[27],"contains":[28],"a":[29,70,89],"subset":[30],"for":[34,80,93],"which":[35],"all":[36,94],"nonzero":[38],"volumes":[39],"${t":[42],"\\choose":[43],"a}$":[44],"subsets":[45],"order":[47],"distinct.":[50],"Beginning":[51],"Erd\u00f6s":[53],"1957,":[55],"function":[57],"$h_{2,d}(n)$":[58,81],"has":[59],"been":[60],"closely":[61],"studied":[62],"is":[64,86],"known":[65,78],"to":[66],"at":[68,87],"least":[69,88],"power":[71,90],"$n$.":[73],"We":[74],"improve":[75],"best":[77],"bound":[79],"show":[83],"$d$.":[97]},"counts_by_year":[{"year":2016,"cited_by_count":1}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}