Parametrized Ramsey theory of infinite block sequences of vectors

We show that the infinite-dimensional versions of Gowers’ ${\mathrm{F}\mathrm{I}\mathrm{N}}_k$ and ${\mathrm{F}\mathrm{I}\mathrm{N}}_{\pm k}$ 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.