Splitting families and complete separability

TL;DR

This paper establishes a key equality between splitting numbers and constructs a completely separable maximal almost disjoint family, advancing understanding in set theory and combinatorics.

Contribution

It proves that al s equals al s_{,} and constructs a completely separable maximal almost disjoint family under certain set-theoretic assumptions.

Findings

Proves al s = al s_{,}.

Constructs a completely separable maximal almost disjoint family.

Partially answers a question of Shelah.

Abstract

We answer a question from Raghavan and Stepr{\=a}ns' paper on weakly tight families by showing that $\mathfrak{s} = {\mathfrak{s}}_{\omega, \omega}$. Then we use this to construct a completely separable maximal almost disjoint family under $\s \leq \a$, partially answering a question of Shelah.