More ZFC inequalities between cardinal invariants

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.