{"id":"https://openalex.org/W4414357675","doi":"https://doi.org/10.1007/s00493-025-00174-y","title":"Weak saturation rank: a failure of the linear algebraic approach to weak saturation","display_name":"Weak saturation rank: a failure of the linear algebraic approach to weak saturation","publication_year":2025,"publication_date":"2025-09-19","ids":{"openalex":"https://openalex.org/W4414357675","doi":"https://doi.org/10.1007/s00493-025-00174-y"},"language":"lv","primary_location":{"id":"doi:10.1007/s00493-025-00174-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00493-025-00174-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00493-025-00174-y.pdf","source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00493-025-00174-y.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5098937155","display_name":"Nikolai Terekhov","orcid":null},"institutions":[{"id":"https://openalex.org/I153845743","display_name":"Moscow Institute of Physics and Technology","ror":"https://ror.org/00v0z9322","country_code":"RU","type":"education","lineage":["https://openalex.org/I153845743"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Nikolai Terekhov","raw_affiliation_strings":["Department of Discrete Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Russia"],"affiliations":[{"raw_affiliation_string":"Department of Discrete Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Russia","institution_ids":["https://openalex.org/I153845743"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5054841023","display_name":"Maksim Zhukovskii","orcid":"https://orcid.org/0000-0001-8763-9533"},"institutions":[{"id":"https://openalex.org/I91136226","display_name":"University of Sheffield","ror":"https://ror.org/05krs5044","country_code":"GB","type":"education","lineage":["https://openalex.org/I91136226"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Maksim Zhukovskii","raw_affiliation_strings":["School of Computer Science, University of Sheffield, S1 4DP, Sheffield, UK"],"affiliations":[{"raw_affiliation_string":"School of Computer Science, University of Sheffield, S1 4DP, Sheffield, UK","institution_ids":["https://openalex.org/I91136226"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5054841023"],"corresponding_institution_ids":["https://openalex.org/I91136226"],"apc_list":null,"apc_paid":null,"fwci":1.4425,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.83118168,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"45","issue":"5","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9607999920845032,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9607999920845032,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.9569000005722046,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9516000151634216,"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/saturation","display_name":"Saturation (graph theory)","score":0.6804999709129333},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5911999940872192},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5511000156402588},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.42890000343322754},{"id":"https://openalex.org/keywords/vector-space","display_name":"Vector space","score":0.3873000144958496},{"id":"https://openalex.org/keywords/algebraic-connectivity","display_name":"Algebraic connectivity","score":0.33489999175071716},{"id":"https://openalex.org/keywords/connectivity","display_name":"Connectivity","score":0.3109000027179718}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8726999759674072},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6841999888420105},{"id":"https://openalex.org/C9930424","wikidata":"https://www.wikidata.org/wiki/Q7426587","display_name":"Saturation (graph theory)","level":2,"score":0.6804999709129333},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5911999940872192},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5511000156402588},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5080999732017517},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.42890000343322754},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.3873000144958496},{"id":"https://openalex.org/C7948225","wikidata":"https://www.wikidata.org/wiki/Q4723998","display_name":"Algebraic connectivity","level":4,"score":0.33489999175071716},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.3109000027179718},{"id":"https://openalex.org/C125080357","wikidata":"https://www.wikidata.org/wiki/Q45715","display_name":"Complete graph","level":3,"score":0.301800012588501},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3012999892234802},{"id":"https://openalex.org/C73586568","wikidata":"https://www.wikidata.org/wiki/Q2600211","display_name":"Parameter space","level":2,"score":0.29899999499320984},{"id":"https://openalex.org/C49766605","wikidata":"https://www.wikidata.org/wiki/Q207643","display_name":"Linear map","level":2,"score":0.28870001435279846},{"id":"https://openalex.org/C103257674","wikidata":"https://www.wikidata.org/wiki/Q6934937","display_name":"Multiple edges","level":3,"score":0.271699994802475},{"id":"https://openalex.org/C176370821","wikidata":"https://www.wikidata.org/wiki/Q1826459","display_name":"Linear space","level":2,"score":0.2662999927997589},{"id":"https://openalex.org/C162307627","wikidata":"https://www.wikidata.org/wiki/Q204833","display_name":"Enhanced Data Rates for GSM Evolution","level":2,"score":0.26460000872612}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00493-025-00174-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00493-025-00174-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00493-025-00174-y.pdf","source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00493-025-00174-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00493-025-00174-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00493-025-00174-y.pdf","source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4414357675.pdf"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W122755046","https://openalex.org/W1842412249","https://openalex.org/W1992614687","https://openalex.org/W1998853144","https://openalex.org/W2020193410","https://openalex.org/W2034962669","https://openalex.org/W2045453971","https://openalex.org/W2045768640","https://openalex.org/W2063472176","https://openalex.org/W2070418505","https://openalex.org/W2073328888","https://openalex.org/W2107074561","https://openalex.org/W2108704847","https://openalex.org/W2148333157","https://openalex.org/W2478722229","https://openalex.org/W2911322315","https://openalex.org/W3097178316","https://openalex.org/W3102723242","https://openalex.org/W3133344740","https://openalex.org/W3184378663","https://openalex.org/W3197336419","https://openalex.org/W3210517074","https://openalex.org/W4406306054"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"Given":[1],"a":[2,6,38,56,63,72,95,278],"graph":[3,39],"F":[4,13,66,141,246],"and":[5,307],"positive":[7],"integer":[8],"n":[9,42],",":[10,142,156,173],"the":[11,32,46,80,290],"weak":[12,89],"-saturation":[14],"number":[15,34],"$${\\textrm{wsat}}(K_n,F)$$":[16],"<mml:math":[17,111,126,149,159,180,215,257],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[18,112,127,150,160,181,216,258],"<mml:mrow>":[19,128,161,182,185,202,217,259],"<mml:mtext>wsat</mml:mtext>":[20,221,263],"<mml:mo>(</mml:mo>":[21,165,189,203,222,264],"<mml:msub>":[22,113,134,151,183,197,223,265],"<mml:mi>K</mml:mi>":[23,114,135,224,266],"<mml:mi>n</mml:mi>":[24,115,136,225,267],"</mml:msub>":[25,116,137,154,196,200,226,268],"<mml:mo>,</mml:mo>":[26,227,269],"<mml:mi>F</mml:mi>":[27,130,167,191,228,270],"<mml:mo>)</mml:mo>":[28,170,194,205,229,271],"</mml:mrow>":[29,138,171,195,206,209,230,272],"</mml:math>":[30,117,139,155,172,210,231,273],"is":[31,294],"minimum":[33],"of":[35,65,79,109,140,239,280],"edges":[36,47,107],"in":[37,49,69],"H":[40,50],"on":[41,88],"vertices":[43],"such":[44,247],"that":[45,59,76,283],"missing":[48],"can":[51],"be":[52,94],"added,":[53],"one":[54,78],"at":[55],"time,":[57],"so":[58],"every":[60,122,250],"edge":[61],"creates":[62],"copy":[64,123],".":[67,118,211,232,274],"Kalai":[68],"1985":[70],"introduced":[71],"linear":[73],"algebraic":[74],"approach":[75,282,293],"became":[77],"most":[81],"efficient":[82],"tools":[83],"to":[84,106,301],"prove":[85,237],"lower":[86,286],"bounds":[87,287],"saturation":[90],"numbers.":[91],"Let":[92],"W":[93,253],"vector":[96,251],"space":[97,252],"spanned":[98],"by":[99],"vectors":[100],"w":[101],"(":[102],"e":[103,108],")":[104],"assigned":[105],"$$K_n$$":[110],"Suppose":[119],"that,":[120,248],"for":[121,249],"$$F'\\subset":[124],"K_n$$":[125],"<mml:msup>":[129,166,190],"<mml:mo>\u2032</mml:mo>":[131,168,192],"</mml:msup>":[132,169,193],"<mml:mo>\u2282</mml:mo>":[133],"there":[143],"exist":[144],"non-zero":[145],"scalars":[146],"$$\\lambda":[147],"_e$$":[148],"<mml:mi>\u03bb</mml:mi>":[152,198],"<mml:mi>e</mml:mi>":[153,162,186,199,204],"$$e\\in":[157],"E(F')$$":[158],"<mml:mo>\u2208</mml:mo>":[163,187],"<mml:mi>E</mml:mi>":[164,188],"satisfying":[174],"$$\\sum":[175],"_{e\\in":[176],"E(F')}\\lambda":[177],"_e":[178],"w(e)=0$$":[179],"<mml:mo>\u2211</mml:mo>":[184],"<mml:mi>w</mml:mi>":[201],"<mml:mo>=</mml:mo>":[207],"<mml:mn>0</mml:mn>":[208],"Then":[212],"$$\\textrm{dim}W\\le":[213],"{\\textrm{wsat}}(K_n,F)$$":[214],"<mml:mtext>dim</mml:mtext>":[218,260],"<mml:mi>W</mml:mi>":[219,261],"<mml:mo>\u2264</mml:mo>":[220],"In":[233],"this":[234,240,281],"paper,":[235],"we":[236,242,297],"limitations":[238],"approach:":[241],"find":[243],"infinitely":[244],"many":[245],"as":[254],"above,":[255],"$$\\textrm{dim}W&lt;{\\textrm{wsat}}(K_n,F)$$":[256],"<mml:mo>&lt;</mml:mo>":[262],"We":[275],"also":[276],"introduce":[277],"modification":[279],"yields":[284],"tight":[285],"even":[288],"when":[289],"original":[291],"direct":[292],"insufficient.":[295],"Finally,":[296],"generalise":[298],"our":[299],"results":[300],"random":[302],"graphs,":[303,306],"complete":[304],"multipartite":[305],"hypergraphs.":[308]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-03-11T06:11:40.159057","created_date":"2025-10-10T00:00:00"}
