Parametrizing the Ramsey theory of vector spaces I: Discrete spaces

TL;DR

This paper explores how the Ramsey theory of block sequences in infinite-dimensional discrete vector spaces can be parametrized by perfect sets, revealing new combinatorial dichotomies and examining ultrafilter preservation under Sacks forcing.

Contribution

It introduces a parametrization of the Ramsey theory in discrete vector spaces and establishes new combinatorial dichotomies for definable families.

Findings

Ramsey theory can be parametrized by perfect sets in this context

Established combinatorial dichotomies for definable families of partitions and linear transformations

Analyzed the preservation of selective ultrafilters under Sacks forcing

Abstract

We show that the Ramsey theory of block sequences in infinite-dimensional discrete vector spaces can be parametrized by perfect sets. As special cases, we prove combinatorial dichotomies for definable families of partitions and linear transformations on those spaces. We also consider the extent to which analogues of selective ultrafilters in this setting are preserved by Sacks forcing.