|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)  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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