Ramsey Property and Block Oscillation Stability on Normalized Sequences in Banach Spaces

TL;DR

This paper explores the connection between Ramsey theory and stability concepts in Banach space sequences, introducing new models and stability notions that generalize spreading models and establish equivalences with Ramsey's theorem.

Contribution

It introduces the notion of block oscillation stability and block asymptotic models, extending the Brunel-Sucheston theorem and linking these concepts to Ramsey's theorem.

Findings

Ramsey theorem is equivalent to the existence of block oscillation stable subsequences.

Introduces $( ext{barrier})$-block asymptotic models as generalizations of spreading models.

Proves Brunel-Sucheston theorem holds for these new models.

Abstract

A well-known application of the Ramsey Theorem in the Banach Space Theory is the proof of the fact that every normalized basic sequence has a subsequence which generates a spreading model (the Brunel-Sucheston Theorem). Based on this application, as an intermediate step, we can talk about the notion of $(k,\varepsilon)-$oscillation stable sequence, which will be described and analyzed more generally in this article. Indeed, we introduce the notion $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequence where $(\mathcal{B}_i)_{i=1}^k$ is a finite sequence of barriers and using what we will call blocks of barriers. In particular, we prove that the Ramsey Theorem is equivalent to the statement ``for every finite sequence $(\mathcal{B}_i)_{i=1}^k$ of barriers, every $\varepsilon>0$ and every normalized sequence $(x_i)_{i\in\mathbb{N}}$ there is a subsequence $(x_i)_{i\in M}$ that is $((\mathcal{B}_i\cap\mathcal{P}(M))_{i=1}^k,\varepsilon)-$block oscillation stable'', where $\mathcal{P}(M)$ is the power set of the infinite set M. Besides, we introduce the $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic model of a normalized basic sequence where $(\mathcal{B}_i)_{i\in\mathbb{N}}$ is a sequence of barriers. These models are a generalization of the spreading models and are related to the $((\mathcal{B}_i)_{i=1}^k,\varepsilon)-$block oscillation stable sequences. We show that the Brunel-Sucheston is satisfied for the $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic models, and we also prove that this result is equivalent to the Ramsey Theorem. The difference between our theorem and the Brunel-Sucheston Theorem is based on the number of different models that are obtained from the same normalized basic sequence through them. This and other observations about $(\mathcal{B}_i)_{i\in\mathbb{N}}-$block asymptotic models are noted in an example at the end of the article.