{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T02:53:22Z","timestamp":1770519202900,"version":"3.49.0"},"reference-count":15,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2021,9,28]],"date-time":"2021-09-28T00:00:00Z","timestamp":1632787200000},"content-version":"vor","delay-in-days":58,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2021,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The two\u2010dimensional version of the classical Mycielski theorem says that for every comeager or conull set  there exists a perfect set  such that . We consider a strengthening of this theorem by replacing a perfect square with a rectangle , where <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic> are bodies of some types of trees with . In particular, we show that for every comeager G<jats:sub>\u03b4<\/jats:sub> set  there exist a Miller tree  and a uniformly perfect tree  such that  and that  cannot be a Miller tree. In the case of measure we show that for every subset <jats:italic>F<\/jats:italic> of  of full measure there exists a uniformly perfect tree  such that  and no side of such a rectangle can be a body of a Silver tree or a Miller tree. We also show some properties of forcing extensions from which we derive nonstandard proofs of Mycielski\u2010like theorems via the Shoenfield Absoluteness\u00a0Theorem.<\/jats:p>","DOI":"10.1002\/malq.202000002","type":"journal-article","created":{"date-parts":[[2021,9,28]],"date-time":"2021-09-28T20:58:29Z","timestamp":1632862709000},"page":"271-281","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Mycielski among trees"],"prefix":"10.1002","volume":"67","author":[{"given":"Marcin","family":"Michalski","sequence":"first","affiliation":[{"name":"Department of Computer Science Faculty of Fundamental Problems of Technology Wroc\u0142aw University of Science and Technology  50\u2010370 Wroc\u0142aw Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert","family":"Ra\u0142owski","sequence":"additional","affiliation":[{"name":"Department of Computer Science Faculty of Fundamental Problems of Technology Wroc\u0142aw University of Science and Technology  50\u2010370 Wroc\u0142aw Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Szymon","family":"\u017beberski","sequence":"additional","affiliation":[{"name":"Department of Computer Science Faculty of Fundamental Problems of Technology Wroc\u0142aw University of Science and Technology  50\u2010370 Wroc\u0142aw Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2021,9,28]]},"reference":[{"key":"e_1_2_6_2_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-2015-12419-7"},{"key":"e_1_2_6_3_1","doi-asserted-by":"publisher","DOI":"10.1201\/9781439863466"},{"key":"e_1_2_6_4_1","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/5.1.108"},{"key":"e_1_2_6_5_1","volume-title":"Set Theory, Millenium Edition, Springer Monographs in Mathematics","author":"Jech T.","year":"2003"},{"key":"e_1_2_6_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"e_1_2_6_7_1","doi-asserted-by":"publisher","DOI":"10.4064\/ba8029-5-2016"},{"key":"e_1_2_6_8_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-02-06584-X"},{"key":"e_1_2_6_9_1","doi-asserted-by":"publisher","DOI":"10.1090\/proc\/13513"},{"key":"e_1_2_6_10_1","doi-asserted-by":"publisher","DOI":"10.4064\/fm-61-2-165-169"},{"key":"e_1_2_6_11_1","first-page":"179","article-title":"Cohen real and disjoint refinement of perfect sets","volume":"41","author":"Repick\u00fd M.","year":"2000","journal-title":"Comment. 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