Parametrizing the Ramsey theory of block sequences I: Discrete vector spaces

Abstract

We show that the Ramsey theory of block sequences in infinite-dimensional vector spaces can be parametrized by perfect sets. This is applied to prove combinatorial dichotomies concerning families of linear transformations and partitions of vector spaces, respectively. We also prove a (necessarily partial) preservation result for analogues of selective ultrafilters in this setting, under Sacks forcing.