# Schauder Bases Having Many Good Block Basic Sequences

## TL;DR

This paper explores the structure of Banach spaces with bases that have many good block basic sequences, establishing strong equivalences and stabilization results related to asymptotic geometric properties.

## Contribution

It introduces new combinatorial conditions characterizing $1$-asymptotic $\, ext{ell}_p$ spaces and proves a stabilization theorem for good block basic sequences.

## Key findings

- Block tree assumptions are equivalent to $1$-asymptotic $\, ext{ell}_p$ spaces.
- Every block basic sequence being good is a stronger condition.
- A stabilization theorem produces a basis with all normalized block sequences being good.

## Abstract

In the study of asymptotic geometry in Banach spaces, a basic sequence which gives rise to a spreading model has been called a good sequence. It is well known that every normalized basic sequence in a Banach space has a subsequence which is good. We investigate the assumption that every normalized block tree relative to a basis has a branch which is good. This combinatorial property turns out to be very strong and is equivalent to the space being $1$-asymptotic $\ell_p$ for some $1\leq p\leq\infty$. We also investigate the even stronger assumption that every block basic sequence of a basis is good. Finally, using the Hindman-Milliken-Taylor theorem, we prove a stabilization theorem which produces a basic sequence all of whose normalized constant coefficient block basic sequences are good, and we present an application of this stabilization.