{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T01:12:36Z","timestamp":1773105156457,"version":"3.50.1"},"reference-count":14,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","funder":[{"name":"NSF","award":["DMS-1703708"],"award-info":[{"award-number":["DMS-1703708"]}]},{"name":"NSF","award":["DMS-1800323"],"award-info":[{"award-number":["DMS-1800323"]}]},{"name":"NSF","award":["DMS-1855757"],"award-info":[{"award-number":["DMS-1855757"]}]},{"name":"NSF","award":["DMS-1945592"],"award-info":[{"award-number":["DMS-1945592"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:p> Assume [Formula: see text]. If [Formula: see text] is an ordinal and X is a set of ordinals, then [Formula: see text] is the collection of order-preserving functions [Formula: see text] which have uniform cofinality [Formula: see text] and discontinuous everywhere. The weak partition properties on [Formula: see text] and [Formula: see text] yield partition measures on [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every [Formula: see text] and function [Formula: see text], there is a club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. For every [Formula: see text] and function [Formula: see text], there is an [Formula: see text]-club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of [Formula: see text]. [Formula: see text]. [Formula: see text]. [Formula: see text]. It will also be shown that [Formula: see text] has the J\u00f3nsson property: For every [Formula: see text], there is an [Formula: see text] with [Formula: see text] so that [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0219061322500295","type":"journal-article","created":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T08:25:23Z","timestamp":1667982323000},"source":"Crossref","is-referenced-by-count":2,"title":["More definable combinatorics around the first and second uncountable cardinals"],"prefix":"10.1142","volume":"23","author":[{"given":"William","family":"Chan","sequence":"first","affiliation":[{"name":"Department of Mathematics. University of North Texas, Denton, TX 76203, USA"}]},{"given":"Stephen","family":"Jackson","sequence":"additional","affiliation":[{"name":"Department of Mathematics. University of North Texas, Denton, TX 76203, USA"}]},{"given":"Nam","family":"Trang","sequence":"additional","affiliation":[{"name":"Department of Mathematics. University of North Texas, Denton, TX 76203, USA"}]}],"member":"219","published-online":{"date-parts":[[2023,1,5]]},"reference":[{"key":"S0219061322500295BIB001","series-title":"Contemporary Mathematics","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1090\/conm\/533\/10510","volume-title":"Set Theory and its Applications","volume":"533","author":"Eduardo Caicedo A.","year":"2011"},{"issue":"2","key":"S0219061322500295BIB002","doi-asserted-by":"crossref","first-page":"1950009","DOI":"10.1142\/S0219061319500090","volume":"19","author":"Chan W.","year":"2019","journal-title":"J. Math. Logic"},{"key":"S0219061322500295BIB003","series-title":"Contemporary Mathematics","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1090\/conm\/752\/15128","volume-title":"Trends in Set Theory","volume":"752","author":"Chan W.","year":"2020"},{"issue":"8","key":"S0219061322500295BIB005","doi-asserted-by":"crossref","first-page":"102988","DOI":"10.1016\/j.apal.2021.102988","volume":"172","author":"Chan W.","year":"2021","journal-title":"Ann. Pure Appl. Logic"},{"issue":"3","key":"S0219061322500295BIB006","doi-asserted-by":"crossref","first-page":"2035","DOI":"10.1090\/tran\/8281","volume":"374","author":"Chan W.","year":"2021","journal-title":"Trans. Amer. Math. Soc"},{"issue":"3","key":"S0219061322500295BIB007","doi-asserted-by":"crossref","first-page":"1725","DOI":"10.1090\/tran\/8573","volume":"375","author":"Chan W.","year":"2022","journal-title":"Trans. Amer. Math. Soc."},{"issue":"4","key":"S0219061322500295BIB009","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1090\/S0894-0347-1990-1057041-5","volume":"3","author":"Harrington L. A.","year":"1990","journal-title":"J. Amer. Math. Soc."},{"issue":"4","key":"S0219061322500295BIB011","doi-asserted-by":"crossref","first-page":"1199","DOI":"10.2307\/2275883","volume":"60","author":"Hjorth G.","year":"1995","journal-title":"J. Symbolic Logic"},{"issue":"4","key":"S0219061322500295BIB012","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1017\/jsl.2014.49","volume":"79","author":"Jackson S.","year":"2014","journal-title":"J. Symb. Log."},{"issue":"670","key":"S0219061322500295BIB013","first-page":"viii+94","volume":"140","author":"Jackson S.","year":"1999","journal-title":"Mem. Amer. Math. Soc."},{"key":"S0219061322500295BIB014","first-page":"1753","author":"Kechris A. S.","year":"2010","journal-title":"Handbook of Set Theory. Vols. 1, 2, 3"},{"key":"S0219061322500295BIB016","doi-asserted-by":"crossref","unstructured":"E. M. Kleinberg,  Infinitary Combinatorics and the Axiom of Determinateness,  Lecture Notes in Mathematics, Vol.  612  (Springer-Verlag,  Berlin, New York,  1977).","DOI":"10.1007\/BFb0069385"},{"issue":"1","key":"S0219061322500295BIB017","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0003-4843(80)90002-9","volume":"18","author":"Silver J. H.","year":"1980","journal-title":"Ann. Math. Logic"},{"issue":"1","key":"S0219061322500295BIB018","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.apal.2005.09.014","volume":"140","author":"Hugh Woodin W.","year":"2006","journal-title":"Ann. Pure Appl. Logic"}],"container-title":["Journal of Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0219061322500295","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T08:24:04Z","timestamp":1687249444000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0219061322500295"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,5]]},"references-count":14,"journal-issue":{"issue":"03","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["10.1142\/S0219061322500295"],"URL":"https:\/\/doi.org\/10.1142\/s0219061322500295","relation":{},"ISSN":["0219-0613","1793-6691"],"issn-type":[{"value":"0219-0613","type":"print"},{"value":"1793-6691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,5]]},"article-number":"2250029"}}