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 ${\mathfrak{s}}_{\theta }\le {\mathfrak{b}}_{\theta }$, 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.

25 pages, to appear