More ZFC inequalities between cardinal invariants

TL;DR

This paper establishes new ZFC theorems and inequalities relating various cardinal invariants on uncountable regular cardinals, expanding understanding of their interrelations without additional set-theoretic assumptions.

Contribution

It provides novel ZFC bounds and equalities among bounding, almost disjointness, reaping, and dominating numbers for uncountable regular cardinals.

Findings

If ()=^+, then _e()=_p()=^+.

Under certain conditions, _g()=^+.

New bounds for () in terms of () and () are established.

Abstract

Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $\kappa=\lambda^+$ for some $\lambda\geq \omega$ and $\mathfrak b(\kappa)=\kappa^+$ then $\mathfrak a_e(\kappa)=\mathfrak a_p(\kappa)=\kappa^+$. If, additionally, $2^{<\lambda}=\lambda$ then $\mathfrak a_g(\kappa)=\kappa^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak d(\kappa)$ in terms of $\mathfrak r(\kappa)$, including $\mathfrak d(\kappa)\leq \mathfrak r_\sigma(\kappa)\leq \mathrm{cof}([\mathfrak r(\kappa)]^\omega)$, and $\mathfrak d(\kappa)\leq \mathfrak r(\kappa)$ whenever $\mathfrak r(\kappa)<\mathfrak b(\kappa)^{+\kappa}$ or $\mathrm{cof}(\mathfrak r(\kappa))\leq \kappa$ holds.