On the bounding, splitting, and distributivity numbers

TL;DR

This paper investigates the relationships among certain cardinal invariants of the power set of natural numbers, demonstrating that all known inequalities can be strict and introducing new bounds and methods to analyze these invariants.

Contribution

It proves that all inequalities among the invariants can be strict and introduces a new upper bound for the bounding number using finite support matrix iterations.

Findings

All inequalities among the invariants can be strict.

A new upper bound for the bounding number $rak h$ is established.

Finite support matrix iterations of ccc posets are utilized to derive results.

Abstract

The cardinal invariants $ \mathfrak h, \mathfrak b, \mathfrak s$ of $\mathcal P (\omega)$ are known to satisfy that $\omega_1 \leq \mathfrak h \leq\min\{\mathfrak b, \mathfrak s\}$. 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}.