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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Determinacy and weakly Ramsey sets in Banach spaces
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by Joan Bagaria and Jordi López-Abad PDF
Trans. Amer. Math. Soc. 354 (2002), 1327-1349 Request permission

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
  • MR Author ID: 340166
  • Email: bagaria@trivium.gh.ub.es
  • Jordi López-Abad
  • Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Spain
  • MR Author ID: 680200
  • Email: abad@mat.uab.es
  • Received by editor(s): March 21, 2000
  • Published electronically: November 21, 2001
  • © Copyright 2001 American Mathematical Society
  • 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
  • MathSciNet review: 1873008