Parameterized partition relations on the real numbers |
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Authors: | Joan Bagaria Carlos A Di Prisco |
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Institution: | 1.ICREA (Institució Catalana de Recerca i Estudis Avan?ats) and Departament de Lògica, Història i Filosofia de la Ciència,Universitat de Barcelona,Barcelona,Spain;2.Departamento de Matemáticas,Instituto Venezolano de Investigaciones Científicas,Caracas,Venezuela |
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Abstract: | We consider several kinds of partition relations on the set of real numbers and its powers, as well as their parameterizations with the set of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness,
that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing
extensions, which yields, for instance, that in these models for every well ordered partition of there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition
of there is and a sequence of perfect sets such that the product lies in one piece of the partition, where is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness.
This work was supported by the research projects MTM 2005-01025 of the Spanish Ministry of Science and Education and 2005SGR-00738
of the Generalitat de Catalunya. A substantial part of the work was carried out while the second-named author was ICREA Visiting
Professor at the Centre de Recerca Matemàtica in Bellaterra (Barcelona), and also during the first-named author’s stays at
the Instituto Venezolano de Investigaciones Científicas and the California Institute of Technology. The authors gratefully
acknowledge the support provided by these institutions. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 03E02 03E35 |
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