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Determinacy and weakly Ramsey sets in Banach spaces


Authors: Joan Bagaria and Jordi López-Abad
Journal: Trans. Amer. Math. Soc. 354 (2002), 1327-1349
MSC (2000): Primary 03E75, 03E15; Secondary 46B45
DOI: https://doi.org/10.1090/S0002-9947-01-02926-9
Published electronically: November 21, 2001
MathSciNet review: 1873008
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Abstract: We give a sufficient condition for a set of block subspaces in an infinite-dimensional Banach space to be weakly Ramsey. Using this condition we prove that in the Levy-collapse of a Mahlo cardinal, every projective set is weakly Ramsey. This, together with a construction of W. H. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. In the case of $c_0$ we prove similar results for a stronger Ramsey property. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. These results are the generalizations to the class of projective sets of some theorems from W. T. Gowers, and our paper ``Weakly Ramsey sets in Banach spaces.''


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Additional Information

Joan Bagaria
Affiliation: Institució Catalana de Recerca i Estudis Avancats (ICREA), 08010 Barcelona, Spain and; Departament de Lògica, Universitat de Barcelona, Baldiri Reixac s/n, 08028 Barcelona, Spain
Email: bagaria@trivium.gh.ub.es

Jordi López-Abad
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Spain
Email: abad@mat.uab.es

DOI: https://doi.org/10.1090/S0002-9947-01-02926-9
Keywords: Weakly Ramsey, block subspaces, projective determinacy
Received by editor(s): March 21, 2000
Published electronically: November 21, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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