{"id":"https://openalex.org/W4207043559","doi":"https://doi.org/10.1017/s0963548321000572","title":"Removal lemmas and approximate homomorphisms","display_name":"Removal lemmas and approximate homomorphisms","publication_year":2022,"publication_date":"2022-01-24","ids":{"openalex":"https://openalex.org/W4207043559","doi":"https://doi.org/10.1017/s0963548321000572"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548321000572","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0963548321000572","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A0618B680623DD624744FA294C9F2504/S0963548321000572a.pdf/div-class-title-removal-lemmas-and-approximate-homomorphisms-div.pdf","source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A0618B680623DD624744FA294C9F2504/S0963548321000572a.pdf/div-class-title-removal-lemmas-and-approximate-homomorphisms-div.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5112361944","display_name":"Jacob Fox","orcid":"https://orcid.org/0000-0002-0664-497X"},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Jacob Fox","raw_affiliation_strings":["Department of Mathematics, Stanford University, Stanford, CA 94305, USA and","Department of Mathematics, Stanford University, Stanford, CA 94305, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Stanford University, Stanford, CA 94305, USA and","institution_ids":["https://openalex.org/I97018004"]},{"raw_affiliation_string":"Department of Mathematics, Stanford University, Stanford, CA 94305, USA","institution_ids":["https://openalex.org/I97018004"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5021133458","display_name":"Yufei Zhao","orcid":"https://orcid.org/0000-0002-1995-3755"},"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":"Yufei Zhao","raw_affiliation_strings":["Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5112361944"],"corresponding_institution_ids":["https://openalex.org/I97018004"],"apc_list":null,"apc_paid":null,"fwci":0.5685,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.55039289,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"31","issue":"4","first_page":"721","last_page":"736"},"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":0.9998999834060669,"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":0.9998999834060669,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9962999820709229,"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":0.9958999752998352,"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/homomorphism","display_name":"Homomorphism","score":0.8735189437866211},{"id":"https://openalex.org/keywords/lemma","display_name":"Lemma (botany)","score":0.8205010890960693},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.767164945602417},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7464504241943359},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5177133083343506},{"id":"https://openalex.org/keywords/graph-homomorphism","display_name":"Graph homomorphism","score":0.5087404847145081},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5034632086753845},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.43116384744644165},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.1771126091480255},{"id":"https://openalex.org/keywords/graph-power","display_name":"Graph power","score":0.16415002942085266},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.050175487995147705}],"concepts":[{"id":"https://openalex.org/C4042151","wikidata":"https://www.wikidata.org/wiki/Q215111","display_name":"Homomorphism","level":2,"score":0.8735189437866211},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.8205010890960693},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.767164945602417},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7464504241943359},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5177133083343506},{"id":"https://openalex.org/C7036158","wikidata":"https://www.wikidata.org/wiki/Q3385162","display_name":"Graph homomorphism","level":5,"score":0.5087404847145081},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5034632086753845},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.43116384744644165},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.1771126091480255},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.16415002942085266},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.050175487995147705},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1017/s0963548321000572","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0963548321000572","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A0618B680623DD624744FA294C9F2504/S0963548321000572a.pdf/div-class-title-removal-lemmas-and-approximate-homomorphisms-div.pdf","source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1017/s0963548321000572","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0963548321000572","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A0618B680623DD624744FA294C9F2504/S0963548321000572a.pdf/div-class-title-removal-lemmas-and-approximate-homomorphisms-div.pdf","source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3103483706","display_name":null,"funder_award_id":"DMS-1764176","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G560730309","display_name":"Graph Theory and Additive Combinatorics","funder_award_id":"1764176","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6894402473","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7926446827","display_name":"Methods in Extremal Combinatorics","funder_award_id":"1855635","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":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4207043559.pdf","grobid_xml":"https://content.openalex.org/works/W4207043559.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W121995582","https://openalex.org/W1836492331","https://openalex.org/W1996392660","https://openalex.org/W2019475553","https://openalex.org/W2023600080","https://openalex.org/W2037773642","https://openalex.org/W2058870462","https://openalex.org/W2072858942","https://openalex.org/W2073279331","https://openalex.org/W2095273580","https://openalex.org/W2124668680","https://openalex.org/W2149054878","https://openalex.org/W2161620758","https://openalex.org/W2164542539","https://openalex.org/W2407637222","https://openalex.org/W2463059379","https://openalex.org/W2499901585","https://openalex.org/W2559529531","https://openalex.org/W2563982062","https://openalex.org/W2763468056","https://openalex.org/W2799132192","https://openalex.org/W2963826098","https://openalex.org/W2964148686","https://openalex.org/W2964194804","https://openalex.org/W2964300648","https://openalex.org/W3099671403","https://openalex.org/W3100735241","https://openalex.org/W3102089577","https://openalex.org/W3102800082","https://openalex.org/W3186616995"],"related_works":["https://openalex.org/W2039793369","https://openalex.org/W830674616","https://openalex.org/W3158819397","https://openalex.org/W4207043559","https://openalex.org/W2169872958","https://openalex.org/W4300410646","https://openalex.org/W2963793606","https://openalex.org/W3045451102","https://openalex.org/W3101124545","https://openalex.org/W2537038771"],"abstract_inverted_index":{"Abstract":[0],"We":[1,107],"study":[2],"quantitative":[3],"relationships":[4],"between":[5],"the":[6,21,89,94,125],"triangle":[7],"removal":[8],"lemma":[9,23,96],"and":[10],"several":[11],"of":[12,73,79,84],"its":[13],"variants.":[14],"One":[15,82],"such":[16,33],"variant,":[17],"which":[18],"we":[19],"call":[20],"triangle-free":[22,36,46,95],",":[24],"states":[25],"that":[26,34,88],"for":[27,113],"each":[28],"$\\epsilon&gt;0$":[29],"there":[30],"exists":[31],"M":[32,52,92],"every":[35],"graph":[37,47],"G":[38,74],"has":[39],"an":[40,55],"$\\epsilon$":[41,56],"-approximate":[42,57],"homomorphism":[43,58],"to":[44,77,129],"a":[45,60],"F":[48,80],"on":[49],"at":[50,68],"most":[51,69],"vertices":[53],"(here":[54],"is":[59,87],"map":[61],"$V(G)":[62],"\\to":[63],"V(F)$":[64],"where":[65,124],"all":[66],"but":[67],"$\\epsilon":[70],"\\left\\lvert{V(G)}\\right\\rvert^2$":[71],"edges":[72,78],"are":[75,127],"mapped":[76],").":[81],"consequence":[83],"our":[85],"results":[86,112],"least":[90],"possible":[91],"in":[93,101,104],"grows":[97],"faster":[98],"than":[99],"exponential":[100],"any":[102],"polynomial":[103],"$\\epsilon^{-1}$":[105],".":[106],"also":[108],"prove":[109],"more":[110],"general":[111],"arbitrary":[114],"graphs,":[115],"as":[116,118],"well":[117],"arithmetic":[119],"analogues":[120],"over":[121],"finite":[122],"fields,":[123],"bounds":[126],"close":[128],"optimal.":[130]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
