A local Ramsey theory for block sequences

TL;DR

This paper develops local Ramsey-theoretic results for block sequences in infinite-dimensional vector spaces, extending classical theorems under large cardinal assumptions and applying to Banach spaces and the Calkin algebra.

Contribution

It introduces new local Ramsey dichotomies for block sequences, extending classical results to broader contexts including Banach spaces and operator algebras.

Findings

Established local Ramsey dichotomies for block sequences.

Extended results to models with large cardinals and $ extbf{L}( ext{R})$-generic filters.

Applied the theory to Banach spaces and the Calkin algebra.

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.