Weakly Ramsey Sets in Banach Spaces

✍ Scribed by Joan Bagaria; Jordi López-Abad

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 347 KB
• Volume: 160
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2001.1983

✦ Synopsis

Gowers' analysis of the combinatorial content of his celebrated dichotomy for infinite-dimensional separable Banach spaces [7] led him to the formulation of the property of being weakly Ramsey applied to sets of block bases, a combinatorial notion related to the classical Ramsey property for infinite sets of positive integers. Let [N] | be the set of all infinite sets of positive integers. With the natural topology induced by the Cantor space via characteristic functions, [N] | is a Polish space. A subset _ [N] | is Ramsey if there exists A # [N] | such that either [A] | _ or [A] | & _=<, where [A] | is the set of all infinite subsets of A. The famous Galvin Prikry theorem asserts that every Borel subset of [N] | is Ramsey ([4]) or, equivalently, that every Borel map from [N] | into a finite space is constant on a cube [A] | .

Silver [17] shows that, in fact, all analytic subsets of [N] | , i.e., the continuous images of Borel sets, are Ramsey. A simpler, more combinatorial proof is given by Ellentuck [3]. The Ramsey property for more complex subsets of [N] | turns out to depend essentially on the axioms of set theory. Thus, for instance, while Go del's axiom of constructibility implies that some continuous image of a co-analytic set is not Ramsey (see [9]), Martin's axiom implies that all such sets are Ramsey [17]. Furthermore, large-cardinal axioms, or determinacy axioms, imply that all projective, or