{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T14:17:07Z","timestamp":1773152227019,"version":"3.50.1"},"reference-count":30,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2004,12]]},"abstract":"<jats:p> We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. <\/jats:p><jats:p> Let the homogeneity number[Formula: see text] of a pair-coloring c:[X]<jats:sup>2<\/jats:sup>\u21922 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2<jats:sup>\u03c9<\/jats:sup>, c<jats:sub> min <\/jats:sub> and c<jats:sub> max <\/jats:sub>, which satisfy [Formula: see text] and prove: <\/jats:p><jats:p> Theorem. (1) For every Polish space X and every continuous pair-coloringc:[X]<jats:sup>2<\/jats:sup>\u21922with[Formula: see text], [Formula: see text] <\/jats:p><jats:p> (2) There is a model of set theory in which[Formula: see text]and[Formula: see text]. <\/jats:p><jats:p> The consistency of [Formula: see text] and of [Formula: see text] follows from [20]. <\/jats:p><jats:p> We prove that [Formula: see text] is equal to the covering number of (2<jats:sup>\u03c9<\/jats:sup>)<jats:sup>2<\/jats:sup> by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c<jats:sub> min <\/jats:sub> gives: <\/jats:p><jats:p> Theorem. There is a model of set theory in which <\/jats:p><jats:p> (1) \u211d<jats:sup>2<\/jats:sup> is coverable by\u2135<jats:sub>1<\/jats:sub>graphs and reflections of graphs of continuous real functions; <\/jats:p><jats:p> (2) \u211d<jats:sup>2<\/jats:sup> is not coverable by\u2135<jats:sub>1<\/jats:sub>graphs and reflections of graphs of Lipschitz real functions. <\/jats:p><jats:p> Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes [Formula: see text] from row (3). <\/jats:p>","DOI":"10.1142\/s0219061304000334","type":"journal-article","created":{"date-parts":[[2005,1,14]],"date-time":"2005-01-14T11:39:54Z","timestamp":1105702794000},"page":"109-145","source":"Crossref","is-referenced-by-count":12,"title":["CONTINUOUS RAMSEY THEORY ON POLISH SPACES AND COVERING THE PLANE BY FUNCTIONS"],"prefix":"10.1142","volume":"04","author":[{"given":"STEFAN","family":"GESCHKE","sequence":"first","affiliation":[{"name":"II. Mathematisches Institut, Freie Universit\u00e4t Berlin, Arnimallee 3, 14195 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MARTIN","family":"GOLDSTERN","sequence":"additional","affiliation":[{"name":"Algebra, TU Wien, Wiedner Hauptstrasse 8-10\/104, A-1040 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MENACHEM","family":"KOJMAN","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-04-07422-2"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(84)90024-1"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1985-0784008-9"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(03)00227-9"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1002\/0471722154.scard"},{"key":"rf6","doi-asserted-by":"crossref","DOI":"10.1201\/9781439863466","volume-title":"Set Theory: On the Structure of the Real Line","author":"Bartoszy\u0144ski T.","year":"1995"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.2307\/2273356"},{"key":"rf8","doi-asserted-by":"crossref","first-page":"101","DOI":"10.4064\/fm-79-2-101-106","volume":"79","author":"Baumgartner J. E.","journal-title":"Fund. Math."},{"key":"rf9","first-page":"1","volume":"87","author":"Baumgartner J. E.","journal-title":"Surveys in Set Theory"},{"key":"rf11","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/s10107-003-0449-8","volume":"97","author":"Chudnovsky M.","journal-title":"Math. Program. B"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.2307\/2371274"},{"key":"rf15","series-title":"General Topology","volume-title":"Sigma Series in Pure Mathematics","author":"Engelking R.","year":"1989"},{"key":"rf16","series-title":"Probabilistic Methods in Combinatorics","volume-title":"Probability and Mathematical Statistics","volume":"7","author":"Erd\u0151s P.","year":"1974"},{"key":"rf17","volume":"148","author":"Farah I.","journal-title":"Mem. Amer. Math. Soc."},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1993-1113695-5"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-02-06437-7"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1007\/BF02785863"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1007\/BF01375552"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(82)90004-3"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-8641(01)00207-3"},{"key":"rf24","doi-asserted-by":"crossref","first-page":"143","DOI":"10.4064\/fm-164-2-143-163","volume":"164","author":"Kojman M.","journal-title":"Fund. Math."},{"key":"rf25","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-51-01882-0"},{"key":"rf26","series-title":"Set Theory. An Introduction to Independence Proofs","volume-title":"Studies in Logic and the Foundations of Mathematics","volume":"102","author":"Kunen K.","year":"1983"},{"key":"rf27","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-02-06376-1"},{"key":"rf28","doi-asserted-by":"publisher","DOI":"10.2307\/1971037"},{"key":"rf30","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.2002.v9.n5.a2"},{"key":"rf31","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-99-02197-2"},{"key":"rf32","series-title":"Partition Problems in Topology","volume-title":"Contemporary Mathematics","volume":"84","author":"Todor\u010devi\u0107 S.","year":"1989"},{"key":"rf33","doi-asserted-by":"publisher","DOI":"10.1016\/0166-8641(93)90127-Y"},{"key":"rf34","volume":"793","author":"Zapletal J.","journal-title":"Mem. Amer. Math. Soc."}],"container-title":["Journal of Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0219061304000334","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T00:30:59Z","timestamp":1565137859000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0219061304000334"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,12]]},"references-count":30,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2011,11,21]]},"published-print":{"date-parts":[[2004,12]]}},"alternative-id":["10.1142\/S0219061304000334"],"URL":"https:\/\/doi.org\/10.1142\/s0219061304000334","relation":{},"ISSN":["0219-0613","1793-6691"],"issn-type":[{"value":"0219-0613","type":"print"},{"value":"1793-6691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,12]]}}}