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A local Ramsey theory for block sequences

Iian B. Smythe

Abstract

We develop local forms of Ramsey-theoretic dichotomies for block sequences in infinite-dimensional vector spaces, analogous to Mathias' selective coideal form of Silver's theorem for analytic partitions of $[\mathbb{N}]^\infty$. Under large cardinals, these results are extended to partitions in $\mathbf{L}(\mathbb{R})$ and $\mathbf{L}(\mathbb{R})$-generic filters of block sequences are characterized. Variants of these results are also established for block sequences in Banach spaces and for projections in the Calkin algebra.

A local Ramsey theory for block sequences

Abstract

We develop local forms of Ramsey-theoretic dichotomies for block sequences in infinite-dimensional vector spaces, analogous to Mathias' selective coideal form of Silver's theorem for analytic partitions of . Under large cardinals, these results are extended to partitions in and -generic filters of block sequences are characterized. Variants of these results are also established for block sequences in Banach spaces and for projections in the Calkin algebra.

Paper Structure

This paper contains 10 sections, 65 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Families of block sequences
  3. Games with vectors and a local Rosendal dichotomy
  4. Stronger properties of families
  5. Constructions of filters in $\mathrm{bb}^\infty(E)$
  6. Connections to filters on a countable set
  7. Extending to universally Baire sets and $\mathbf{L}(\mathbb{R})$
  8. Normed spaces and a local Gowers dichotomy
  9. Projections in the Calkin algebra
  10. Further questions

Key Result

Theorem 1

Let $B$ be an infinite-dimensional Banach space with a Schauder basis. If $\mathbb{A}$ is an analytic set of normalized block sequences, then for any $\Delta>0$, there is a block sequence $Y$ such that either

Theorems & Definitions (140)

  • Theorem : Gowers MR1421876 MR1954235
  • Theorem : Rosendal MR2604856
  • Theorem : Silver MR0332480
  • Theorem : Mathias MR0491197
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • ...and 130 more