Authors: Lopez-Abad, Jordi
Todorčević, Stevo 
Title: Positional graphs and conditional structure of weakly null sequences
Journal: Advances in Mathematics
Volume: 242
First page: 163
Last page: 186
Issue Date: 1-Aug-2013
Rank: M21a
ISSN: 0001-8708
DOI: 10.1016/j.aim.2013.04.010
Abstract: 
We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer n there are normalized weakly-null sequences of length ωn without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that ωω is the minimal cardinal κ that could possibly have the property that every weakly null κ-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κ which is smaller than the first ω-Erdos cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either c0 or lp, with p≥1.
Keywords: Banach problem | Minimal walks | Non-separable Banach spaces | Polarized Ramsey | Separable quotient problem | Unconditional and subsymmetric basic sequences
Publisher: Elsevier
Project: Ministerio de Economía y Competitividad, Project MTM2012-31286

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