Call a subset of $\mathbf{ FIN}_k$ small if it does not contain a copy of $\langle{ A\rangle}$ for some infinite block sequence $A \in \mathbf{ FIN}_k^{ [\infty]}$. Gowers' $\mathbf{ FIN}_k$ theorem asserts that the set of small subsets of $\mathbf{ FIN}_k$ forms an ideal, so it is sensible to consider almost disjoint families of $\mathbf{ FIN}_k$ with respect to the ideal of small subsets of $\mathbf{ FIN}_k$. We shall show that $\mathfrak{ a}_{ \mathbf{ FIN}_k}$, the smallest possible cardinality of an infinite mad family of $\mathbf{ FIN}_k$, is uncountable.
Categories: math.LO math.CO