{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T00:18:48Z","timestamp":1648599528736},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":3754,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2003,12]]},"abstract":"<jats:p>The theme of this paper is the generalization of theorems about partitions of the sets of points and lines of finite-dimensional Euclidean spaces \u211d<jats:sup><jats:italic>d<\/jats:italic><\/jats:sup>to vector spaces over \u211d of arbitrary dimension and, more generally still, to arbitrary vector spaces over other fields so long as these fields are not too big. These theorems have their origins in the following striking theorem of Sierpi\u0144ski [12] which appeared a half century ago.<\/jats:p><jats:p>Sierpi\u0144ski's Theorem.<jats:italic>The Continuum Hypothesis is equivalent to: There is a partition<\/jats:italic>{<jats:italic>X, Y, Z<\/jats:italic>}<jats:italic>of<\/jats:italic>\u211d<jats:sup>3<\/jats:sup><jats:italic>such that if \u2113 is a line parallel to the x-axis<\/jats:italic>[respectively:<jats:italic>y-axis, z-axis<\/jats:italic>]<jats:italic>then X<\/jats:italic>\u2229<jats:italic>\u2113<\/jats:italic>[respectively:<jats:italic>Y<\/jats:italic>\u2229<jats:italic>\u2113, Z<\/jats:italic>\u2229<jats:italic>\u2113<\/jats:italic>]<jats:italic>is finite<\/jats:italic>.<\/jats:p><jats:p>The history of this theorem and some of its subsequent developments are discussed in the very interesting article by Simms [13]. Sierpi\u0144ski's Theorem was generalized by Kuratowski [9] to partitions of \u211d<jats:sup><jats:italic>n<\/jats:italic>+2<\/jats:sup>into<jats:italic>n<\/jats:italic>+ 2 sets obtaining an equivalence with<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200008276_inline1\" \/>. The geometric character that Sierpi\u0144ski's Theorem and its generalization by Kuratowski appear to have is bogus, since the lines parallel to coordinate axes are essentially combinatorial, rather than geometric, objects. The following version of Kuratowski's theorem emphasizes its combinatorial character.<\/jats:p><jats:p>Kuratowski's Theorem.<jats:italic>Let n &lt; \u03c9 and A be any set. Then<\/jats:italic>\u2223<jats:italic>A<\/jats:italic>\u2223 \u2264 \u2135<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub><jats:italic>if and only if there is a partition P<\/jats:italic>:<jats:italic>A<\/jats:italic><jats:sup><jats:italic>n<\/jats:italic>+2<\/jats:sup>\u2192<jats:italic>n<\/jats:italic>+ 2<jats:italic>such that if i<\/jats:italic>\u2264<jats:italic>n<\/jats:italic>+ 1<jats:italic>and \u2113 is a line parallel to the i-th coordinate axis, then<\/jats:italic>{<jats:italic>x<\/jats:italic>\u2208<jats:italic>\u2113<\/jats:italic>:<jats:italic>P<\/jats:italic>(<jats:italic>x<\/jats:italic>) =<jats:italic>i<\/jats:italic>}<jats:italic>is finite<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1067620179","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:15:24Z","timestamp":1109798124000},"page":"1171-1180","source":"Crossref","is-referenced-by-count":0,"title":["Partitioning large vector spaces"],"prefix":"10.1017","volume":"68","author":[{"given":"James H.","family":"Schmerl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200008276_ref005","unstructured":"Galvin F. , an apparently unpublished result from [7]."},{"key":"S0022481200008276_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0166-8641(84)90049-X"},{"key":"S0022481200008276_ref004","first-page":"51","article-title":"Some remarks on set theory, III","volume":"2","author":"Erd\u0151s","year":"1954","journal-title":"Michigan Mathematics Journal"},{"key":"S0022481200008276_ref002","doi-asserted-by":"publisher","DOI":"10.1017\/S030500410000195X"},{"key":"S0022481200008276_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58043-7_13"},{"key":"S0022481200008276_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BF02843714"},{"key":"S0022481200008276_ref010","doi-asserted-by":"crossref","first-page":"183","DOI":"10.4064\/fm-160-2-183-196","article-title":"Countable partitions of the sets of points and lines","volume":"160","author":"Schmerl","year":"1999","journal-title":"Fundamenta Mathematicae"},{"key":"S0022481200008276_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100037105"},{"key":"S0022481200008276_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(01)00042-2"},{"key":"S0022481200008276_ref009","doi-asserted-by":"publisher","DOI":"10.4064\/fm-38-1-14-17"},{"key":"S0022481200008276_ref011","volume-title":"Fundamenta Mathematicae","author":"Schmerl"},{"key":"S0022481200008276_ref013","first-page":"69","article-title":"Sierpi\u0144ski's theorem","volume":"65","author":"Simms","year":"1991","journal-title":"Bulletin of the Belgian Mathematical Society, Simon Stevin"},{"key":"S0022481200008276_ref003","doi-asserted-by":"publisher","DOI":"10.4064\/fm-52-3-277-281"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200008276","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T06:15:55Z","timestamp":1625552155000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200008276\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,12]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2003,12]]}},"alternative-id":["S0022481200008276"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1067620179","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,12]]}}}