arXiv 0807.22052008-07-14math.FAmath.LO

An exact Ramsey principle for block sequences

Christian Rosendal

英文摘要：

We prove an exact, i.e., formulated without $Δ$-expansions, Ramsey principle for infinite block sequences in vector spaces over countable fields, where the two sides of the dichotomic principle are represented by respectively winning strategies in Gowers' block sequence game and winning strategies in the infinite asymptotic game. This allows us to recover Gowers' dichotomy theorem for block sequences in normed vector spaces by a simple application of the basic determinacy theorem for infinite asymptotic games.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Parametrized Ramsey theory of infinite block sequences of vectors

We show that the infinite-dimensional versions of Gowers' $\mathrm{FIN}_k$ and $\mathrm{FIN}_{\pm k}$ theorems can be parametrized by an infinite sequence of perfect subsets of $2^ω$. To do so, we use ultra-Ramsey theory to obtain exact and approximate versions of a result which combines elements from both Gowers' theorems and the Hales-Jewett theorem. As a consequence, we obtain a parametrized version of Gowers' $c_0$ theorem.

2020-06-18math.COmath.FAmath.LO

A new approach to the Ramsey-type games and the Gowers dichotomy in F-spaces

We give a new approach to the Ramsey-type results of Gowers on block bases in Banach spaces and apply our results to prove the Gowers dichotomy in F-spaces.

2008-05-31math.FA

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

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…

2020-09-07math.COmath.FA

Local Ramsey Spaces in Matet Forcing Extensions and Finitely Many Near-Coherence Classes

We introduce Gowers--Matet forcing with a finite sequence of pairwise non-isomorphic Ramsey ultrafilters over $ω$, and with this forcing we settle the long-standing problem of the spectrum of numbers near-coherence classes. We prove that for any finite $n \geq 1$, there is a forcing extension with exactly $n$ near-coherence classes of ultrafilters. For evaluating the new forcing, we prove a strengthening of Gowers's theorem on colourings of ${\rm Fin}_k$.

2019-07-30math.FAmath.LO

Determinacy of adversarial Gowers games

We prove a game theoretic dichotomy for $G_{δσ}$ sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis' proof of $G_{δσ}$ determinacy.

2012-08-11math.FAmath.LO

On Banach Spaces containing $l_p$ or $c_0$

We use the Gowers block Ramsey theorem to characterize Banach spaces containing isomorphs of $\ell_p$ (for some $1 \leq p < \infty$) or $c_0$.

2008-10-02math.FA

更多相关论文收起更多论文

Uniform upper estimates and the repeated averages hierarchy

We use the repeated averages hierarchy to prove a Ramsey theorem regarding uniform upper estimates of convex block sequences of weakly null sequences. The base case of the theorem recovers a result of Freeman.

2020-03-25math.FA

A discretized approach to W.T. Gowers' game

We give an alternative proof of W. T. Gowers' theorem on block bases by reducing it to a discrete analogue on specific countable nets. We also give a Ramsey type result on k-tuples of block sequences in a normed linear space with a Schauder basis.

2010-03-01math.COmath.FA

On asymptotic models in Banach spaces

A well known application of Ramsey's Theorem to Banach Space Theory is the notion of a spreading model (e'_i) of a normalized basic sequence (x_i) in a Banach space X. We show how to generalize the construction to define a new creature (e_i), which we call an asymptotic model of X. Every spreading model of X is an asymptotic model of X and in most settings, such as if X is reflexive, every normalized block basis of an asymptotic model is itself an asymptotic model. We also show how to use the Hindman-Milliken Theorem--a strengthened form of Ramsey's Theorem--to generate asymptotic models with …

2001-10-14math.FA