Homogeneous families on trees and subsymmetric basic sequences

TL;DR

This paper extends Tsirelson's space to larger index sets, exploring density conditions in Banach spaces that influence the existence of subsymmetric basic sequences, and constructs spaces without such sequences below the first Mahlo cardinal.

Contribution

It introduces a method to analyze Banach spaces using homogeneous families on trees, extending Tsirelson's space to larger densities and demonstrating the non-existence of subsymmetric sequences under certain conditions.

Findings

Existence of reflexive Banach spaces of density κ without subsymmetric basic sequences for κ below the first Mahlo cardinal.

Development of a framework using homogeneous families on trees to analyze the asymptotic structure of Banach spaces.

Inductive construction of collections of families with controlled complexity on large index sets.

Abstract

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 $\kappa$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $\kappa$ 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.