Parametrized Ramsey theory of infinite block sequences of vectors

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.