{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T10:17:15Z","timestamp":1773224235486,"version":"3.50.1"},"reference-count":22,"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":466,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2012,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000116X_inline01\"\/> is an <jats:italic>n-grid<\/jats:italic> if each <jats:italic>E<jats:sub>i<\/jats:sub><\/jats:italic>, is an equivalence relation on <jats:italic>A<\/jats:italic> and whenever <jats:italic>X<\/jats:italic> and <jats:italic>Y<\/jats:italic> are equivalence classes of, respectively, distinct <jats:italic>E<jats:sub>i<\/jats:sub><\/jats:italic>, and <jats:italic>E<jats:sub>j<\/jats:sub><\/jats:italic>, then <jats:italic>X<\/jats:italic> \u2229 <jats:italic>Y<\/jats:italic> is finite. A coloring <jats:italic>\u03c7<\/jats:italic>: <jats:italic>A<\/jats:italic> \u2192 <jats:italic>n<\/jats:italic> is <jats:italic>acceptable<\/jats:italic> if whenever <jats:italic>X<\/jats:italic> is an equivalence class of <jats:italic>E<jats:sub>i<\/jats:sub><\/jats:italic>, then {<jats:italic>x<\/jats:italic> \u2208 <jats:italic>X<\/jats:italic>: <jats:italic>\u03c7<\/jats:italic>(<jats:italic>x<\/jats:italic>) = <jats:italic>i<\/jats:italic>} is finite. If <jats:italic>B<\/jats:italic> is any set, then the <jats:italic>n<\/jats:italic>-cube <jats:italic>B<jats:sup>n<\/jats:sup><\/jats:italic> = (<jats:italic>B<jats:sup>n<\/jats:sup><\/jats:italic>; <jats:italic>E<\/jats:italic><jats:sub>0<\/jats:sub>, \u2026, <jats:italic>E<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub>) is considered as an <jats:italic>n<\/jats:italic>-grid, where the equivalence classes of <jats:italic>E<jats:sub>i<\/jats:sub><\/jats:italic> are the lines parallel to the <jats:italic>i<\/jats:italic>-th coordinate axis. Kuratowski [9], generalizing the <jats:italic>n<\/jats:italic> = 3 case proved by Sierpihski [17], proved that \u211d<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup> has an acceptable coloring iff 2<jats:sup>\u2135<jats:sub>0<\/jats:sub><\/jats:sup> \u2264 \u2135<jats:sub><jats:italic>n<\/jats:italic>\u22122<\/jats:sub>. The main result is: if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000116X_inline02\"\/> is a semialgebraic (i.e., first-order definable in the field of reals) <jats:italic>n<\/jats:italic>-grid, then the following are equivalent: (1) if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000116X_inline02\"\/> embeds all finite <jats:italic>n<\/jats:italic>-cubes, then 2<jats:sup>\u2135<jats:sub>0<\/jats:sub><\/jats:sup> \u2264 \u2135<jats:sub><jats:italic>n<\/jats:italic>\u22122<\/jats:sub>: (2) if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000116X_inline02\"\/> embeds \u211d<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>, then 2<jats:sup>\u2135<jats:sub>0<\/jats:sub><\/jats:sup> \u2264 \u2135<jats:sub><jats:italic>n<\/jats:italic>\u22122<\/jats:sub>; (3) <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120000116X_inline02\"\/> has an acceptable coloring.<\/jats:p>","DOI":"10.2178\/jsl.7704060","type":"journal-article","created":{"date-parts":[[2012,11,2]],"date-time":"2012-11-02T15:23:52Z","timestamp":1351869832000},"page":"1165-1183","source":"Crossref","is-referenced-by-count":1,"title":["A generalization of Sierpi\u0144ski's paradoxical decompositions: Coloring semialgebraic grids"],"prefix":"10.1017","volume":"77","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":"S002248120000116X_ref011","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100074612"},{"key":"S002248120000116X_ref009","doi-asserted-by":"publisher","DOI":"10.4064\/fm-38-1-14-17"},{"key":"S002248120000116X_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(01)00042-2"},{"key":"S002248120000116X_ref006","first-page":"101","article-title":"On partitions of lines and space","volume":"145","author":"Erd\u0151s","year":"1994","journal-title":"Fundamenta Mathematicae"},{"key":"S002248120000116X_ref014","first-page":"1171","volume":"68","author":"Schmerl","year":"2003","journal-title":"Partitioning large vector spaces"},{"key":"S002248120000116X_ref005","doi-asserted-by":"publisher","DOI":"10.4064\/fm203-1-6"},{"key":"S002248120000116X_ref004","unstructured":"de la Vega Ramiro , Coloring grids, preprint."},{"key":"S002248120000116X_ref013","doi-asserted-by":"publisher","DOI":"10.4064\/fm177-3-2"},{"key":"S002248120000116X_ref016","first-page":"1","volume-title":"Bull. 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