On the bounding, splitting, and distributivity numbers

Abstract

The cardinal invariants $\mathfrak{h},\mathfrak{b},\mathfrak{s}$ of $\mathcal{P}(\ddot{I}‰)$ are known to satisfy that $\ddot{I}{‰}_1\le \mathfrak{h}\le \min \{\mathfrak{b},\mathfrak{s}\}$. We prove that all inequalities can be strict. We also introduce a new upper bound for $\mathfrak{h}$ and show that it can be less than $\mathfrak{s}$. The key method is to utilize finite support matrix iterations of ccc posets following \cite{BlassShelah}.