Authors: Brech, Christina
Lopez-Abad, Jorge
Todorčević, Stevo 
Title: Homogeneous families on trees and subsymmetric basic sequences
Journal: Advances in Mathematics
Volume: 334
First page: 322
Last page: 388
Issue Date: 20-Aug-2018
Rank: M21a
ISSN: 0001-8708
DOI: 10.1016/j.aim.2018.06.008
We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal κ smaller than the first Mahlo cardinal there is a reflexive Banach space of density κ without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic structure of the spaces. The collections of families are of independent interest and their existence is proved inductively. The fundamental stepping up argument is the analysis of such collections of families on trees.
Keywords: Families of finite sets | Nonseparable Banach spaces | Spreading models
Publisher: Elsevier
Project: FAPESP, Grants 2012/24463-7, 2015/26654-2 and 2013/24827-1
CNPq, Grants 307942/2012-0 and 454112/2015-7
Ministerio de Economía y Competitividad, Grant MTM2012-31286
USP-COFECUB, Grant 31466UC
NSERC, Grant 455916
CNRS, Grant UMR7586

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