Abstract
We show that the infinite-dimensional versions of Gowers' FINk and FIN±k theorems can be parametrized by an infinite sequence of perfect subsets of 2ω. 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' c0 theorem.
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