{"id":"https://openalex.org/W1659233256","doi":"https://doi.org/10.37236/241","title":"On the Size of Minimal Unsatisfiable Formulas","display_name":"On the Size of Minimal Unsatisfiable Formulas","publication_year":2009,"publication_date":"2009-01-30","ids":{"openalex":"https://openalex.org/W1659233256","doi":"https://doi.org/10.37236/241","mag":"1659233256"},"language":"en","primary_location":{"id":"doi:10.37236/241","is_oa":true,"landing_page_url":"https://doi.org/10.37236/241","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1n3/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1n3/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5061128187","display_name":"Choongbum Lee","orcid":null},"institutions":[{"id":"https://openalex.org/I161318765","display_name":"University of California, Los Angeles","ror":"https://ror.org/046rm7j60","country_code":"US","type":"education","lineage":["https://openalex.org/I161318765"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Choongbum Lee","raw_affiliation_strings":["Department of Mathematics, UCLA, Los Angeles, CA, 90095"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, UCLA, Los Angeles, CA, 90095","institution_ids":["https://openalex.org/I161318765"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5061128187"],"corresponding_institution_ids":["https://openalex.org/I161318765"],"apc_list":null,"apc_paid":null,"fwci":0.3535,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.58904425,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"16","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9983000159263611,"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"}},"topics":[{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9983000159263611,"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/T11596","display_name":"Constraint Satisfaction and Optimization","score":0.9936000108718872,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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":0.9878000020980835,"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/combinatorics","display_name":"Combinatorics","score":0.780762255191803},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7205502986907959},{"id":"https://openalex.org/keywords/hypergraph","display_name":"Hypergraph","score":0.6864223480224609},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.5393388867378235},{"id":"https://openalex.org/keywords/chromatic-scale","display_name":"Chromatic scale","score":0.417097806930542},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.37034547328948975},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13762399554252625}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.780762255191803},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7205502986907959},{"id":"https://openalex.org/C2781221856","wikidata":"https://www.wikidata.org/wiki/Q840247","display_name":"Hypergraph","level":2,"score":0.6864223480224609},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.5393388867378235},{"id":"https://openalex.org/C196956537","wikidata":"https://www.wikidata.org/wiki/Q202021","display_name":"Chromatic scale","level":2,"score":0.417097806930542},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37034547328948975},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13762399554252625},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/241","is_oa":true,"landing_page_url":"https://doi.org/10.37236/241","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1n3/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.214.2475","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.214.2475","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/Volume_16/PDF/v16i1n3.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.245.7678","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.245.7678","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/0811.0427v1.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/241","is_oa":true,"landing_page_url":"https://doi.org/10.37236/241","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1n3/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320332195","display_name":"Samsung","ror":"https://ror.org/04w3jy968"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1659233256.pdf","grobid_xml":"https://content.openalex.org/works/W1659233256.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W2015211698","https://openalex.org/W2025484786","https://openalex.org/W2031823362","https://openalex.org/W4249436157"],"related_works":["https://openalex.org/W4376608589","https://openalex.org/W3138003926","https://openalex.org/W4300037846","https://openalex.org/W1630514295","https://openalex.org/W3094179932","https://openalex.org/W3005801850","https://openalex.org/W2950595062","https://openalex.org/W1524556636","https://openalex.org/W1988533413","https://openalex.org/W3127534791"],"abstract_inverted_index":{"An":[0],"unsatisfiable":[1,20],"formula":[2],"is":[3],"called":[4],"minimal":[5,19],"if":[6],"it":[7],"becomes":[8],"satisfiable":[9],"whenever":[10],"any":[11],"of":[12,35,44],"its":[13],"clauses":[14,25],"are":[15],"removed.":[16],"We":[17],"construct":[18],"$k$-SAT":[21],"formulas":[22],"with":[23],"$\\Omega(n^k)$":[24],"for":[26],"$k":[27],"\\geq":[28],"3$,":[29],"thereby":[30],"negatively":[31],"answering":[32],"a":[33,56],"question":[34],"Rosenfeld.":[36],"This":[37],"should":[38],"be":[39],"compared":[40],"to":[41],"the":[42],"result":[43],"Lov\u00e1sz":[45],"[Studia":[46],"Scientiarum":[47],"Mathematicarum":[48],"Hungarica":[49],"11,":[50],"1974,":[51],"p113-114]":[52],"which":[53],"asserts":[54],"that":[55],"critically":[57],"3-chromatic":[58],"$k$-uniform":[59],"hypergraph":[60],"can":[61],"have":[62],"at":[63],"most":[64],"${n\\choose":[65],"k-1}$":[66],"edges.":[67]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}