MAD Families of Gowers' Infinite Block Sequences

Call a subset of FINₖ small if it does not contain a copy of P for some infinite block sequence P ∈ FINₖ[∞]. Gowers' FINₖ theorem asserts that the set of small subsets of FINₖ forms an ideal, so it is sensible to consider almost disjoint families of FINₖ with respect to the ideal of small subsets of FINₖ. We shall show that aFINₖ, the smallest possible cardinality of an infinite mad family of FINₖ, is uncountable.