{"id":"https://openalex.org/W2074195064","doi":"https://doi.org/10.1145/2421096.2421115","title":"Proving expansion in three steps","display_name":"Proving expansion in three steps","publication_year":2012,"publication_date":"2012-08-27","ids":{"openalex":"https://openalex.org/W2074195064","doi":"https://doi.org/10.1145/2421096.2421115","mag":"2074195064"},"language":"en","primary_location":{"id":"doi:10.1145/2421096.2421115","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2421096.2421115","pdf_url":null,"source":{"id":"https://openalex.org/S4210206003","display_name":"ACM SIGACT News","issn_l":"0163-5700","issn":["0163-5700","1943-5827"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM SIGACT News","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041051779","display_name":"Amir Yehudayoff","orcid":"https://orcid.org/0000-0002-0177-1814"},"institutions":[{"id":"https://openalex.org/I174306211","display_name":"Technion \u2013 Israel Institute of Technology","ror":"https://ror.org/03qryx823","country_code":"IL","type":"education","lineage":["https://openalex.org/I174306211"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Amir Yehudayoff","raw_affiliation_strings":["Technion-IIT, Haifa, Israel","Technion (IIT), Haifa, Israel"],"affiliations":[{"raw_affiliation_string":"Technion-IIT, Haifa, Israel","institution_ids":["https://openalex.org/I174306211"]},{"raw_affiliation_string":"Technion (IIT), Haifa, Israel","institution_ids":["https://openalex.org/I174306211"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5041051779"],"corresponding_institution_ids":["https://openalex.org/I174306211"],"apc_list":null,"apc_paid":null,"fwci":0.4926,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.61405599,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"43","issue":"3","first_page":"67","last_page":"84"},"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":0.9988999962806702,"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":0.9988999962806702,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9965999722480774,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9957000017166138,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/chess-endgame","display_name":"Chess endgame","score":0.7666114568710327},{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.6878994703292847},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.5556215047836304},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.49284958839416504},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.48601680994033813},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.44726884365081787},{"id":"https://openalex.org/keywords/mixing","display_name":"Mixing (physics)","score":0.4358074963092804},{"id":"https://openalex.org/keywords/terminology","display_name":"Terminology","score":0.43572238087654114},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4238757789134979},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3641563951969147},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.1662355661392212}],"concepts":[{"id":"https://openalex.org/C118491383","wikidata":"https://www.wikidata.org/wiki/Q332225","display_name":"Chess endgame","level":2,"score":0.7666114568710327},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.6878994703292847},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.5556215047836304},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49284958839416504},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.48601680994033813},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.44726884365081787},{"id":"https://openalex.org/C138777275","wikidata":"https://www.wikidata.org/wiki/Q6884054","display_name":"Mixing (physics)","level":2,"score":0.4358074963092804},{"id":"https://openalex.org/C547195049","wikidata":"https://www.wikidata.org/wiki/Q1725664","display_name":"Terminology","level":2,"score":0.43572238087654114},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4238757789134979},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3641563951969147},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1662355661392212},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/2421096.2421115","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2421096.2421115","pdf_url":null,"source":{"id":"https://openalex.org/S4210206003","display_name":"ACM SIGACT News","issn_l":"0163-5700","issn":["0163-5700","1943-5827"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM SIGACT News","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W1506462246","https://openalex.org/W1530740238","https://openalex.org/W1596185498","https://openalex.org/W1705784892","https://openalex.org/W1822888486","https://openalex.org/W2000713788","https://openalex.org/W2001753575","https://openalex.org/W2018047324","https://openalex.org/W2026391697","https://openalex.org/W2031808668","https://openalex.org/W2045377861","https://openalex.org/W2048958388","https://openalex.org/W2068801641","https://openalex.org/W2069084184","https://openalex.org/W2293527178","https://openalex.org/W2912205456","https://openalex.org/W2949278404","https://openalex.org/W4235946010"],"related_works":["https://openalex.org/W54267352","https://openalex.org/W2786801901","https://openalex.org/W2955468987","https://openalex.org/W1517387034","https://openalex.org/W2002491919","https://openalex.org/W2536541347","https://openalex.org/W23628338","https://openalex.org/W2108021684","https://openalex.org/W140462016","https://openalex.org/W2003745232"],"abstract_inverted_index":{"This":[0],"text":[1],"is":[2,27,68,76,83],"meant":[3],"to":[4,8,51,69,84],"be":[5],"an":[6],"introduction":[7],"a":[9,18,56,86,98,109],"recent":[10],"strategy":[11,26,41],"introduced":[12],"by":[13],"Bourgain":[14],"and":[15,61,115],"Gamburd":[16],"(following":[17],"work":[19],"of":[20,43,55,74,91,96],"Helfgott)":[21],"for":[22,29,89,101,107],"proving":[23,108],"graph-expansion.":[24],"The":[25,40,93],"designed":[28],"graphs":[30],"H":[31,75],"that":[32,71],"are":[33,104],"defined":[34],"using":[35,112,116],"some":[36],"underlying":[37],"group":[38],"G.":[39,92,102],"consists":[42,95],"three":[44,53],"steps,":[45],"which,":[46],"in":[47],"Sarnak's":[48],"terminology,":[49],"correspond":[50],"the":[52,64,66,72,79,81],"steps":[54],"chess":[57],"game:":[58],"opening,":[59,65],"middle-game":[60],"endgame.":[62],"In":[63,78],"objective":[67],"prove":[70,85],"girth":[73],"logarithmic.":[77],"middle-game,":[80],"goal":[82],"productgrowth":[87],"theorem":[88],"subsets":[90],"endgame":[94],"establishing":[97],"\"mixing":[99],"property\"":[100],"There":[103],"two":[105],"methods":[106],"mixing":[110],"property:":[111],"pairwise":[113],"independence":[114],"basic":[117],"representation":[118],"theory.":[119]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2013,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
