{"id":"https://openalex.org/W1994382042","doi":"https://doi.org/10.2307/2272412","title":"Weak compactness and square bracket partition relations","display_name":"Weak compactness and square bracket partition relations","publication_year":1972,"publication_date":"1972-12-01","ids":{"openalex":"https://openalex.org/W1994382042","doi":"https://doi.org/10.2307/2272412","mag":"1994382042"},"language":"en","primary_location":{"id":"doi:10.2307/2272412","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272412","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","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/A5034562840","display_name":"E. M. Kleinberg","orcid":null},"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":true,"raw_author_name":"E. M. Kleinberg","raw_affiliation_strings":["Massachusetts Institute of Technology, Cambridge, Massachusetts 02139","Massachusetts Institute of Technology"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, Cambridge, Massachusetts 02139","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041739806","display_name":"Richard A. Shore","orcid":"https://orcid.org/0000-0003-0381-5259"},"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":"R. A. Shore","raw_affiliation_strings":["Massachusetts Institute of Technology, Cambridge, Massachusetts 02139","Massachusetts Institute of Technology"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, Cambridge, Massachusetts 02139","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5034562840"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":0.6514,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.64083399,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"37","issue":"4","first_page":"673","last_page":"676"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9864000082015991,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9858999848365784,"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/compact-space","display_name":"Compact space","score":0.8365732431411743},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.7927923202514648},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7879098653793335},{"id":"https://openalex.org/keywords/equivalence-relation","display_name":"Equivalence relation","score":0.752777099609375},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6563735008239746},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4350703954696655},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.24916508793830872}],"concepts":[{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.8365732431411743},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.7927923202514648},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7879098653793335},{"id":"https://openalex.org/C156103551","wikidata":"https://www.wikidata.org/wiki/Q130998","display_name":"Equivalence relation","level":2,"score":0.752777099609375},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6563735008239746},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4350703954696655},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.24916508793830872}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2272412","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272412","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jsl/1183738448","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183738448","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W2026216307","https://openalex.org/W2044949519","https://openalex.org/W2086478638"],"related_works":["https://openalex.org/W2121543990","https://openalex.org/W2402800040","https://openalex.org/W2000020320","https://openalex.org/W4252969132","https://openalex.org/W1981078319","https://openalex.org/W2008917324","https://openalex.org/W2581541046","https://openalex.org/W4387315838","https://openalex.org/W2805799113","https://openalex.org/W2077709279"],"abstract_inverted_index":{"Although":[0],"there":[1,89],"are":[2],"many":[3],"characterizations":[4],"of":[5,12,30,51,73,83,119,133,141,151,157,169,209,232],"weakly":[6],"compact":[7],"cardinals":[8,43],"(e.g.":[9],"in":[10,28,167],"terms":[11,29,168],"indescnbability":[13],"and":[14,41,44,54,107],"tree":[15,212],"properties":[16],"as":[17,19,57],"well":[18],"compactness)":[20],"the":[21,48,79,120,129,138,148,177,181,188,192,199,207,211,230],"most":[22],"interesting":[23],"set-theoretic":[24],"(combinatorial)":[25],"one":[26,214],"is":[27,122,126,137,147,165,202,221,229],"partition":[31,49,72,171,234],"relations.":[32,235],"To":[33],"be":[34],"more":[35,241],"precise":[36],"we":[37],"define":[38],"for":[39,98,205,224],"\u03ba":[40,65,75,84,95,106],"\u03b1":[42,69,87,114],"n":[45,67,77,80,112,130,226],"an":[46],"integer":[47],"relation":[50,178,189],"Erd\u00f6s,":[52],"Hajnal":[53],"Rado":[55],"[2]":[56],"follows:":[58],"For":[59],"every":[60,225],"function":[61],"F":[62,99,108,125],":":[63],"[":[64,74],"]":[66,76,111],"\u2192":[68],"(called":[70,96],"a":[71,91],",":[78,85,115,145],"-element":[81,131],"subsets":[82,132],"into":[86],"pieces),":[88],"exists":[90],"set":[92],"C":[93,104,110,134],"\u2286":[94],"homogeneous":[97],")":[100],"such":[101],"that":[102,146,217],"card":[103],"=":[105],"\u2033[":[109],"\u2260":[113],"i.e.":[116,144],"some":[117],"element":[118],"range":[121],"omitted":[123],"when":[124,162],"restricted":[127],"to":[128,175,186,201,210],".":[135,194,227],"It":[136],"simplest":[139],"(nontrivial)":[140],"these":[142,170,233],"relations,":[143],"well-known":[149],"equivalent":[150,222],"weak":[152,163],"compactness.":[153],"1":[154],"Two":[155],"directions":[156],"inquiry":[158],"immediately":[159],"suggest":[160],"themselves":[161],"compactness":[164],"described":[166],"relations:":[172],"(a)":[173],"Trying":[174],"strengthen":[176],"by":[179,190],"increasing":[180,191],"superscript\u2014e.g.,":[182],"\u2014and":[183],"(b)":[184],"trying":[185],"weaken":[187],"subscript\u2014e.g.,":[193],"As":[195],"it":[196],"turns":[197],"out,":[198],"strengthening":[200],"only":[203],"illusory":[204],"using":[206],"equivalence":[208],"property":[213],"quickly":[215],"sees":[216],"implies":[218],"(and":[219],"so":[220],"to)":[223],"Thus":[228],"strongest":[231],"The":[236],"second":[237],"question":[238],"seems":[239],"much":[240],"difficult.":[242]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}