On configurations concerning cardinal characteristics at regular cardinals

Abstract

We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak{s}_\theta,\mathfrak{p}_\theta,\mathfrak{g}_\theta,\mathfrak{r}_\theta,\mathfrak{t}_\theta$ at uncountable regular cardinals $\theta$. Motivated by a theorem of Raghavan-Shelah who proved that $\mathfrak{s}_\theta\leq\mathfrak{b}_\theta$, we explore in the first part of the paper the consistency of inequalities comparing $\mathfrak{s}_\theta$ with $\mathfrak{p}_\theta$ and $\mathfrak{g}_\theta$. In the second part of the paper we study variations of the extender-based Radin forcing to establish several consistency results concerning $\mathfrak{r}_\theta$ from hyper-measurability assumptions, results which were previously known to be consistent only from supercompactness assumptions. In doing so, we answer several questions which appeared in the literature.