Parametrizing the Ramsey theory of vector spaces I: Discrete spaces

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.

arXiv comment37 pages. Accepted for publication in Ann. Pure Appl. Logic. Major revisions from previous versions include removing section and results on "destroying" strategic filters, whose proofs contained an error, replaced with preservation results under large cardinal hypotheses, thanks largely to the referee. Other minor corrections and edits made throughout