Mad families of Gowers' infinite block sequences

Abstract

Call a subset of ${\mathbf{FIN}}_k$ small if it does not contain a copy of $⟨A⟩$ 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.

9 pages. 18 Mar 25: This version provides a more streamlined proof of Theorem 1.1