{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T06:19:09Z","timestamp":1648880349644},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2021,8,12]],"date-time":"2021-08-12T00:00:00Z","timestamp":1628726400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2021,9]]},"abstract":"Abstract<\/jats:title>We make comments on some problems Erd\u0151s and Hajnal posed in their famous problem list. Let X<\/jats:italic> be a graph on \n$\\omega _1$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with the property that every uncountable set A<\/jats:italic> of vertices contains a finite set s<\/jats:italic> such that each element of \n$A-s$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is joined to one of the elements of s<\/jats:italic>. Does then X<\/jats:italic> contain an uncountable clique? (Problem 69) We prove that both the statement and its negation are consistent. Do there exist circuitfree graphs \n$\\{X_n:n<\\omega \\}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> on \n$\\omega _1$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that if \n$A\\in [\\omega _1]^{\\aleph _1}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, then \n$\\{n<\\omega :X_n\\cap [A]^2=\\emptyset \\}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is finite? (Problem 61) We show that the answer is yes under CH, and no under Martin\u2019s axiom. Does there exist \n$F:[\\omega _1]^2\\to 3$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with all three colors appearing in every uncountable set, and with no triangle of three colors. (Problem 68) We give a different proof of Todorcevic\u2019 theorem that the existence of a \n$\\kappa $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-Suslin tree gives \n$F:[\\kappa ]^2\\to \\kappa $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> establishing \n$\\kappa \\not \\to [\\kappa ]^2_{\\kappa }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with no three-colored triangles. This statement in turn implies the existence of a \n$\\kappa $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-Aronszajn tree.<\/jats:p>","DOI":"10.1017\/jsl.2021.56","type":"journal-article","created":{"date-parts":[[2021,8,12]],"date-time":"2021-08-12T02:55:04Z","timestamp":1628736904000},"page":"1116-1123","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["NOTES ON SOME ERD\u0150S\u2013HAJNAL PROBLEMS"],"prefix":"10.1017","volume":"86","author":[{"given":"P\u00c9TER","family":"KOMJ\u00c1TH","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,8,12]]},"reference":[{"key":"S0022481221000566_r5","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(81)90005-X"},{"key":"S0022481221000566_r1","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0280381"},{"key":"S0022481221000566_r4","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760522"},{"key":"S0022481221000566_r2","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/025\/0357122"},{"key":"S0022481221000566_r6","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1983-0716846-0"},{"key":"S0022481221000566_r7","first-page":"21","article-title":"Mitchell: Aronszajn trees and the independence of the transfer property","volume":"5","author":"William","year":"1972\/1973","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0022481221000566_r3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02756751"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481221000566","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,9]],"date-time":"2022-03-09T11:13:18Z","timestamp":1646824398000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481221000566\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,12]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["S0022481221000566"],"URL":"https:\/\/doi.org\/10.1017\/jsl.2021.56","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,12]]},"assertion":[{"value":"\u00a9 Association for Symbolic Logic 2021","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}