# On configurations concerning cardinal characteristics at regular cardinals

## TL;DR

This paper investigates the consistency of various configurations of uncountable regular cardinal characteristics, using advanced forcing techniques to establish new results under weaker large cardinal assumptions.

## Contribution

It introduces new consistency results for cardinal characteristics at regular cardinals, reducing the large cardinal assumptions needed and answering open questions in the literature.

## Key findings

- Established the consistency of inequalities between $rak{s}_ heta$, $rak{p}_ heta$, and $rak{g}_ heta$.
- Developed variations of extender-based Radin forcing for $rak{r}_ heta$ under hyper-measurability.
- Provided new results that previously required supercompactness assumptions.

## 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.