On the bounding, splitting, and distributivity numbers

The cardinal invariants $\mathfrak{h},\mathfrak{b},\mathfrak{s}$ of $\mathcal P (\omega)$areknowntosatisfythat$\omega_1 \leq \mathfrak h \leq\min{\mathfrak b, \mathfrak s}$. 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}.