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Using this approach, we build a model in which a very strong form of the negation of Club Guessing at [Formula: see text] known as [Formula: see text] holds together with [Formula: see text], thereby answering a well-known question of Moore. This construction can be described as a finite-support weak forcing iteration with side conditions consisting of suitable graphs of sets of models with markers. The [Formula: see text]-preservation is accomplished through the imposition of copying constraints on the information carried by the condition, as dictated by the edges in the graph. <\/jats:p>","DOI":"10.1142\/s0219061323500095","type":"journal-article","created":{"date-parts":[[2023,5,26]],"date-time":"2023-05-26T09:35:32Z","timestamp":1685093732000},"source":"Crossref","is-referenced-by-count":0,"title":["Few new reals"],"prefix":"10.1142","volume":"24","author":[{"given":"David","family":"Asper\u00f3","sequence":"first","affiliation":[{"name":"School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK"}]},{"given":"Miguel Angel","family":"Mota","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Instituto Tecnol\u00f3gico Aut\u00f3nomo de M\u00e9xico (ITAM), 01080 Mexico City, Mexico"}]}],"member":"219","published-online":{"date-parts":[[2023,6,29]]},"reference":[{"doi-asserted-by":"publisher","key":"S0219061323500095BIB001","DOI":"10.1007\/978-1-4020-5764-9_6"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB002","DOI":"10.4064\/fm-152-2-165-181"},{"key":"S0219061323500095BIB003","series-title":"Tributes","first-page":"5","volume-title":"Infinity, Computability, and Metamathematics: Festschrift Celebrating the 60th Birthdays of Peter Koepke and Philip Welch","volume":"23","author":"Asper\u00f3 D.","year":"2014"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB005","DOI":"10.4064\/fm781-9-2019"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB006","DOI":"10.1007\/s11511-013-0089-7"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB007","DOI":"10.1090\/S0002-9947-2015-06205-9"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB008","DOI":"10.1007\/s11856-015-1250-0"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB009","DOI":"10.1016\/j.apal.2015.12.003"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB010","DOI":"10.1007\/BFb0065979"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB011","DOI":"10.1007\/BF02762012"},{"key":"S0219061323500095BIB012","series-title":"London Mathematical Society Lecture Note Series","volume-title":"Appalachian Set Theory 2006\u20132012","author":"Eisworth T.","year":"2013"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB013","DOI":"10.1090\/S0002-9947-99-02407-1"},{"volume-title":"Set Theory: The Third Millennium Edition, Revised and Expanded","year":"2002","author":"Jech T.","key":"S0219061323500095BIB015"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB016","DOI":"10.1142\/S0219061315500051"},{"volume-title":"Set Theory, an Introduction to Independence Proofs","year":"1980","author":"Kunen K.","key":"S0219061323500095BIB017"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB018","DOI":"10.1090\/proc\/13133"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB019","DOI":"10.1307\/mmj\/1187647002"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB020","DOI":"10.1007\/s11511-013-0092-z"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB021","DOI":"10.1090\/conm\/690\/13871"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB022","DOI":"10.1007\/978-3-662-12831-2"},{"doi-asserted-by":"publisher","key":"S0219061323500095BIB024","DOI":"10.1090\/conm\/031\/763902"}],"container-title":["Journal of Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0219061323500095","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,6]],"date-time":"2024-05-06T02:25:59Z","timestamp":1714962359000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0219061323500095"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,29]]},"references-count":21,"journal-issue":{"issue":"02","published-print":{"date-parts":[[2024,8]]}},"alternative-id":["10.1142\/S0219061323500095"],"URL":"https:\/\/doi.org\/10.1142\/s0219061323500095","relation":{},"ISSN":["0219-0613","1793-6691"],"issn-type":[{"type":"print","value":"0219-0613"},{"type":"electronic","value":"1793-6691"}],"subject":[],"published":{"date-parts":[[2023,6,29]]},"article-number":"2350009"}}