On the bounding, splitting, and distributivity numbers

The cardinal invariants $\mathfrak h, \mathfrak b, \mathfrak s$ of $\mathcal P (ω)$ are known to satisfy that $ω_1 \leq \mathfrak h \leq\min\{\mathfrak b, \mathfrak s\}$. 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}.

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