Mad families of Gowers' infinite block sequences

TL;DR

The paper proves that the smallest size of an infinite mad family in the Gowers FIN$_k$ framework, $\mathfrak{a}_{\mathbf{FIN}_k}$, is uncountable for every $k\ge1$. It achieves this by introducing a tail-controlled invariant in the form of a function $f:\mathbf{FIN}_k\to\mathbb{N}$ with two key properties that tie almost disjointness to bounded $f$ on intersections and enable a diagonal construction to extend any countable ad family. This yields a construction of a new block sequence that is almost disjoint from every member of the given countable family, hence the family cannot be maximal. The result generalizes the uncountability phenomenon from $k=1$ to all $k\ge1$, highlighting a concrete combinatorial invariant as the obstruction to maximality in the FIN$_k$ setting.

Abstract

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.

Theorems & Definitions (18)

• Theorem 1.1: Gowers $\bf{FIN}_k$ Theorem
• Definition 1.2
• Example 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• Lemma 2.3
• proof
• Lemma 2.4