{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T03:12:51Z","timestamp":1649041971288},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":9323,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1988,9]]},"abstract":"<jats:p>In this paper the theory of the core model <jats:italic>K<\/jats:italic> is applied to study certain combinatorial principles. These principles concern the existence of families of almost disjoint functions. The first, the transversal hypothesis, is defined as follows.<\/jats:p><jats:p>Definition. The <jats:italic>transversal hypothesis<\/jats:italic> for <jats:italic>\u03ba<\/jats:italic>, T(<jats:italic>\u03ba<\/jats:italic>), is the following assertion:<\/jats:p><jats:p>There is a sequence \u3008<jats:italic>f<jats:sub>\u03bd<\/jats:sub><\/jats:italic>: <jats:italic>\u03bd<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>\u3009 such that<\/jats:p><jats:p>(a) <jats:italic>f<jats:sub>\u03bd<\/jats:sub><\/jats:italic>: <jats:italic>\u03ba<\/jats:italic> \u2192 <jats:italic>\u03ba<\/jats:italic> regressively for <jats:italic>\u03bd<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>, and<\/jats:p><jats:p>(b) if <jats:italic>\u03bd<\/jats:italic> &lt; <jats:italic>\u03be<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>, then there is <jats:italic>\u03b3<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic> such that <jats:italic>f<jats:sub>\u03bd<\/jats:sub><\/jats:italic>(<jats:italic>\u03b1<\/jats:italic>) \u2260 <jats:italic>f<jats:sub>\u03be<\/jats:sub><\/jats:italic>(<jats:italic>\u03b1<\/jats:italic>) whenever <jats:italic>\u03b3<\/jats:italic> &lt; <jats:italic>\u03b1<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic>.<\/jats:p><jats:p>T(<jats:italic>\u03ba<\/jats:italic>) is a simple consequence of the <jats:italic>Kurepa hypothesis<\/jats:italic> for <jats:italic>\u03ba<\/jats:italic>, i.e. the assertion, KH(<jats:italic>\u03ba<\/jats:italic>), that there is a family <jats:italic>F<\/jats:italic> \u2282 P(<jats:italic>\u03ba<\/jats:italic>) such that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200042092_inline1\" \/> and card({<jats:italic>X<\/jats:italic> \u2229 <jats:italic>\u03b1<\/jats:italic>: <jats:italic>X<\/jats:italic> \u03f5 <jats:italic>F<\/jats:italic>}) \u2264 <jats:italic>\u03b1<\/jats:italic> for <jats:italic>\u03c9<\/jats:italic> &lt; <jats:italic>\u03b1<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic>.<\/jats:p><jats:p>The second principle to be studied, the weak Kurepa hypothesis, is a statement of strength intermediate between the Kurepa and transversal hypotheses.<\/jats:p><jats:p>Definition. The <jats:italic>weak Kurepa hypothesis<\/jats:italic> for <jats:italic>\u03ba<\/jats:italic>, wKH(<jats:italic>\u03ba<\/jats:italic>), is the following assertion:<\/jats:p><jats:p>There is a sequence \u3008<jats:italic>b<jats:sub>\u03bd<\/jats:sub><\/jats:italic>: <jats:italic>\u03bd<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>\u3009 such that<\/jats:p><jats:p>(a) <jats:italic>b<jats:sub>\u03bd<\/jats:sub><\/jats:italic> \u2282 <jats:italic>\u03ba<\/jats:italic> for <jats:italic>\u03bd<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>, and<\/jats:p><jats:p>(b) for each limit <jats:italic>\u03bb<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic> there is <jats:italic>F<jats:sub>\u03bb<\/jats:sub><\/jats:italic>: {<jats:italic>b<jats:sub>\u03bd<\/jats:sub><\/jats:italic> \u2229 <jats:italic>\u03bb<\/jats:italic>: <jats:italic>\u03bd<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>} \u2192 <jats:italic>\u03bb<\/jats:italic> such that setting <jats:italic>f<jats:sub>\u03bd<\/jats:sub><\/jats:italic>(<jats:italic>\u03bb<\/jats:italic>) = <jats:italic>F<jats:sub>\u03bb<\/jats:sub><\/jats:italic>(<jats:italic>b<jats:sub>\u03bd<\/jats:sub><\/jats:italic> \u2229 <jats:italic>\u03bb<\/jats:italic>) for <jats:italic>\u03bd<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup> and limit <jats:italic>\u03bb<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic>, if <jats:italic>\u03bd<\/jats:italic> &lt; <jats:italic>\u03be<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup> there is <jats:italic>\u03b3<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic> such that <jats:italic>f<jats:sub>\u03bd<\/jats:sub><\/jats:italic>(<jats:italic>\u03bb<\/jats:italic>) \u2260 <jats:italic>f<jats:sub>\u03be<\/jats:sub><\/jats:italic>(<jats:italic>\u03bb<\/jats:italic>) whenever <jats:italic>\u03b3<\/jats:italic> &lt; <jats:italic>\u03bb<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic> and <jats:italic>\u03bb<\/jats:italic> is a limit.<\/jats:p>","DOI":"10.2307\/2274577","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:27:13Z","timestamp":1146954433000},"page":"854-877","source":"Crossref","is-referenced-by-count":0,"title":["On the transversal hypothesis and the weak kurepa hypothesis"],"prefix":"10.1017","volume":"53","author":[{"given":"D. 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