{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T15:20:32Z","timestamp":1648740032557},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2012,3,12]],"date-time":"2012-03-12T00:00:00Z","timestamp":1331510400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2012,5]]},"abstract":"<jats:p>A tree <jats:italic>T<\/jats:italic> is said to be homogeneous if it is uniquely rooted and there exists an integer <jats:italic>b<\/jats:italic> \u2265 2, called the branching number of <jats:italic>T<\/jats:italic>, such that every <jats:italic>t<\/jats:italic> \u2208 <jats:italic>T<\/jats:italic> has exactly <jats:italic>b<\/jats:italic> immediate successors. We study the behaviour of measurable events in probability spaces indexed by homogeneous trees.<\/jats:p><jats:p>Precisely, we show that for every integer <jats:italic>b<\/jats:italic> \u2265 2 and every integer <jats:italic>n<\/jats:italic> \u2265 1 there exists an integer <jats:italic>q<\/jats:italic>(<jats:italic>b<\/jats:italic>,<jats:italic>n<\/jats:italic>) with the following property. If <jats:italic>T<\/jats:italic> is a homogeneous tree with branching number <jats:italic>b<\/jats:italic> and {<jats:italic>A<jats:sub>t<\/jats:sub><\/jats:italic>:<jats:italic>t<\/jats:italic> \u2208 <jats:italic>T<\/jats:italic>} is a family of measurable events in a probability space (\u03a9,\u03a3,\u03bc) satisfying \u03bc(<jats:italic>A<jats:sub>t<\/jats:sub><\/jats:italic>)\u2265\u03f5&gt;0 for every <jats:italic>t<\/jats:italic> \u2208 <jats:italic>T<\/jats:italic>, then for every 0&lt;\u03b8&lt;\u03f5 there exists a strong subtree <jats:italic>S<\/jats:italic> of <jats:italic>T<\/jats:italic> of infinite height, such that for every finite subset <jats:italic>F<\/jats:italic> of <jats:italic>S<\/jats:italic> of cardinality <jats:italic>n<\/jats:italic> \u2265 1 we have\n<jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548312000053_eqnU1\" \/><\/jats:disp-formula>\nIn fact, we can take <jats:italic>q<\/jats:italic>(<jats:italic>b<\/jats:italic>,<jats:italic>n<\/jats:italic>)= ((2<jats:sup><jats:italic>b<\/jats:italic><\/jats:sup>\u22121)<jats:sup>2<jats:italic>n<\/jats:italic>\u22121<\/jats:sup>\u22121)\u00b7(2<jats:sup><jats:italic>b<\/jats:italic><\/jats:sup>\u22122)<jats:sup>\u22121<\/jats:sup>. A finite version of this result is also obtained.<\/jats:p>","DOI":"10.1017\/s0963548312000053","type":"journal-article","created":{"date-parts":[[2012,3,12]],"date-time":"2012-03-12T09:57:57Z","timestamp":1331546277000},"page":"374-411","source":"Crossref","is-referenced-by-count":2,"title":["Measurable Events Indexed by Trees"],"prefix":"10.1017","volume":"21","author":[{"given":"PANDELIS","family":"DODOS","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"VASSILIS","family":"KANELLOPOULOS","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"KONSTANTINOS","family":"TYROS","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2012,3,12]]},"reference":[{"key":"S0963548312000053_ref15","volume-title":"Annals of Mathematics Studies","author":"Todorcevic","year":"2010"},{"key":"S0963548312000053_ref12","volume-title":"Oxford Logic Guides","author":"Rose","year":"1984"},{"key":"S0963548312000053_ref10","unstructured":"[10] Polymath D. H. J. (2009) A new proof of the density Hales\u2013Jewett theorem. Preprint, available at http:\/\/arxiv.org\/abs\/0910.3926."},{"key":"S0963548312000053_ref11","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s2-30.1.264"},{"key":"S0963548312000053_ref6","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1963-0143712-1"},{"key":"S0963548312000053_ref1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1981-0609665-0"},{"key":"S0963548312000053_ref2","unstructured":"[2] Dodos P. , Kanellopoulos V. and Karagiannis N. (2010) A density version of the Halpern\u2013L\u00e4uchli theorem. Preprint, available at http:\/\/arxiv.org\/abs\/1006.2671."},{"key":"S0963548312000053_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/BF03041066"},{"key":"S0963548312000053_ref9","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1981-0590416-8"},{"key":"S0963548312000053_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(88)90109-4"},{"key":"S0963548312000053_ref13","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1988-0929498-X"},{"key":"S0963548312000053_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2010.12.002"},{"key":"S0963548312000053_ref7","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1966-0200172-2"},{"key":"S0963548312000053_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(79)90101-8"},{"key":"S0963548312000053_ref5","first-page":"660","article-title":"Partition theorems for the real line","volume":"15","author":"Galvin","year":"1968","journal-title":"Notices Amer. Math. Soc."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548312000053","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,25]],"date-time":"2019-04-25T22:03:32Z","timestamp":1556229812000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548312000053\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3,12]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,5]]}},"alternative-id":["S0963548312000053"],"URL":"https:\/\/doi.org\/10.1017\/s0963548312000053","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,3,12]]}}}