Determinacy and weakly Ramsey sets in Banach spaces

Abstract

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."

A local Ramsey theory for block sequences

“…In §7 we show that, under large cardinal hypothesis, strategic (p + )-filters have complete combinatorics for infinite block sequences with the block subsequence ordering, and generalize Theorem 1.1 to partitions in L(R) (the corresponding extension of Gowers' original result is due to López-Abad [28], see also [6]). This requires an analysis of a Mathias-like notion of forcing used to build generic block sequences.…”

“…5 Every subset of bb ∞ (E) in L(R) is strategically Ramsey. 6 Proof. We follow the proof of Theorem 4 in [28].…”

“…We use supercompactness due to its central role in the literature and verbal brevity. 6 Noé de Rancourt has announced a different proof of this result using methods inspired by determinacy considerations.…”

A Discretized Approach to W. T. Gowers' Game

“…Introduction W. T. Gowers in [11] (see also [10] and [12]) proved a fundamental Ramsey-type theorem for block bases in Banach spaces which led to important discoveries in the geometry of Banach spaces. By now there are several approaches to Gowers' theorem (see [1,2,3,4,14,21]. Also in [7,15,18] there are direct proofs of Gowers' dichotomy and in [6,8,19,22,24] extensions and further applications).…”

Θεωρία Ramsey και εφαρμογές στη θεωρία χώρων Banach

“…Μέχρι στιγμής υπάρχει πληθώρα προσεγγίσεων στο θεώρημα αυτό (βλ. [3,6,7,8,31,44]. Επίσης στα [14,34,40] υπάρχουν αποδείξεις της διχοτομίας του Gowers και στα [13,15,41,45] επεκτάσεις του και περαιτέρω εφαρμογές αυτού).…”