Parametrized Ramsey theory of infinite block sequences of vectors

TL;DR

This paper extends Gowers' and Hales-Jewett theorems to infinite-dimensional block sequences of vectors using ultra-Ramsey theory, introducing parametrized versions that incorporate perfect subsets of the Cantor space.

Contribution

It introduces a novel parametrization of infinite-dimensional Gowers' theorems using ultra-Ramsey theory, combining elements from multiple classical theorems.

Findings

Parametrized infinite-dimensional Gowers' theorems established.

Exact and approximate versions derived via ultra-Ramsey theory.

A parametrized version of Gowers' c_0 theorem obtained.

Abstract

We show that the infinite-dimensional versions of Gowers' $\mathrm{FIN}_k$ and $\mathrm{FIN}_{\pm k}$ theorems can be parametrized by an infinite sequence of perfect subsets of $2^\omega$. To do so, we use ultra-Ramsey theory to obtain exact and approximate versions of a result which combines elements from both Gowers' theorems and the Hales-Jewett theorem. As a consequence, we obtain a parametrized version of Gowers' $c_0$ theorem.