{"id":"https://openalex.org/W7171486071","doi":"https://doi.org/10.48550/arxiv.2607.23500","title":"Formalizing Flag Algebras in Lean","display_name":"Formalizing Flag Algebras in Lean","publication_year":2026,"publication_date":"2026-07-26","ids":{"openalex":"https://openalex.org/W7171486071","doi":"https://doi.org/10.48550/arxiv.2607.23500"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2607.23500","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.23500","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,"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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2607.23500","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5139895232","display_name":"Gyeongwon Jeong","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jeong, Gyeongwon","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5122278367","display_name":"Seonghun Park","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Park, Seonghun","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073753291","display_name":"Jihoon Hyun","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hyun, Jihoon","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5143855605","display_name":"Sang-il Oum","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Oum, Sang-il","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5143747815","display_name":"Hongseok Yang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yang, Hongseok","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.6452999711036682,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.6452999711036682,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.16599999368190765,"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":0.0348999984562397,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.546999990940094},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4544999897480011},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.4259999990463257},{"id":"https://openalex.org/keywords/certificate","display_name":"Certificate","score":0.3862000107765198},{"id":"https://openalex.org/keywords/compiler","display_name":"Compiler","score":0.3614000082015991},{"id":"https://openalex.org/keywords/pentagon","display_name":"Pentagon","score":0.36070001125335693},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.35910001397132874}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6179999709129333},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.546999990940094},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4544999897480011},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.428600013256073},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.4259999990463257},{"id":"https://openalex.org/C96865113","wikidata":"https://www.wikidata.org/wiki/Q2946816","display_name":"Certificate","level":2,"score":0.3862000107765198},{"id":"https://openalex.org/C169590947","wikidata":"https://www.wikidata.org/wiki/Q47506","display_name":"Compiler","level":2,"score":0.3614000082015991},{"id":"https://openalex.org/C2779965921","wikidata":"https://www.wikidata.org/wiki/Q127840","display_name":"Pentagon","level":2,"score":0.36070001125335693},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.35910001397132874},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.30570000410079956},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.29030001163482666},{"id":"https://openalex.org/C74640108","wikidata":"https://www.wikidata.org/wiki/Q1372900","display_name":"Propositional formula","level":5,"score":0.2768999934196472},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.26759999990463257},{"id":"https://openalex.org/C206880738","wikidata":"https://www.wikidata.org/wiki/Q431667","display_name":"Automated theorem proving","level":2,"score":0.25380000472068787},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2524000108242035},{"id":"https://openalex.org/C16221470","wikidata":"https://www.wikidata.org/wiki/Q846564","display_name":"Conjunctive normal form","level":2,"score":0.25119999051094055}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2607.23500","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.23500","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,"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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2607.23500","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.23500","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,"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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Razborov's":[0],"flag":[1,217],"algebra":[2,75,218],"method":[3,35],"is":[4],"a":[5,22,30,42,152,201,212,247],"powerful":[6],"tool":[7],"for":[8,36,155,161],"proving":[9],"asymptotic":[10],"inequalities":[11,193,224],"in":[12,70],"extremal":[13],"graph":[14,209,228],"theory,":[15],"often":[16],"reducing":[17],"the":[18,34,60,63,73,85,96,111,126,148,169,173,178,186,216,220,237,243,252],"task":[19],"to":[20,89],"finding":[21],"finite":[23,37],"certificate":[24,49],"by":[25,55],"semidefinite":[26,98],"programming.":[27],"We":[28,235],"present":[29,251],"machine-checked":[31],"formalization":[32,58],"of":[33,62,76,130,139,168,182,194,204,207],"simple":[38],"graphs,":[39,67,72,157],"together":[40],"with":[41,230],"certificate-to-proof":[43],"compiler":[44,94],"that":[45,176],"turns":[46],"externally":[47],"generated":[48],"data":[50,103],"into":[51,215],"algebraic":[52,127],"proofs":[53,138],"checked":[54],"Lean.":[56],"The":[57,93],"covers":[59],"foundations":[61],"method:":[64],"partially":[65],"labeled":[66],"their":[68],"densities":[69,181],"large":[71],"quotient":[74],"density":[77,113],"expressions,":[78],"graph-limit":[79],"semantics":[80,198],"through":[81],"positive":[82,118],"homomorphisms,":[83],"and":[84,114,123,147,158,164,185,190],"downward":[86],"operators":[87],"used":[88],"average":[90],"out":[91,125],"labels.":[92],"treats":[95],"external":[97],"programming":[99],"output":[100],"as":[101],"candidate":[102],"rather":[104],"than":[105],"trusted":[106],"input:":[107],"Lean":[108],"independently":[109],"computes":[110],"required":[112],"multiplication":[115],"facts,":[116],"verifies":[117],"semidefiniteness":[119],"exactly":[120],"over":[121],"$\\mathbb{Q}$,":[122],"carries":[124],"normalization":[128],"steps":[129],"flag-algebra":[131],"proofs.":[132],"Our":[133,196],"case":[134],"studies":[135],"yield":[136],"formal":[137],"seven":[140],"Tur\u00e1n-type":[141],"upper":[142],"bounds,":[143],"including":[144],"Mantel's":[145,183],"theorem":[146,184],"Erd\u0151s":[149,187],"pentagon":[150,188],"theorem,":[151,189],"$C_4$-density":[153],"bound":[154],"triangle-free":[156],"edge-density":[159],"bounds":[160],"$K_4$-free,":[162],"$K_5$-free,":[163],"$C_5$-free":[165],"graphs.":[166],"Independently":[167],"compiler,":[170],"we":[171],"formalize":[172],"matching":[174],"constructions":[175],"complete":[177,253],"exact":[179],"Tur\u00e1n":[180],"prove":[191],"two":[192,205,244],"Goodman.":[195],"constrained":[197,227],"also":[199],"prompted":[200],"meta-theoretic":[202],"comparison":[203],"ways":[206],"imposing":[208],"constraints:":[210],"building":[211],"hereditary":[213],"constraint":[214],"from":[219],"start,":[221],"or":[222],"testing":[223],"afterward":[225],"on":[226],"limits":[229],"labels":[231],"chosen":[232],"at":[233],"random.":[234],"state":[236],"resulting":[238],"root-plantability":[239],"criterion":[240],"characterizing":[241],"when":[242],"approaches":[245],"agree;":[246],"forthcoming":[248],"paper":[249],"will":[250],"account.":[254]},"counts_by_year":[],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2026-07-29T00:00:00"}
