# Lower consistency bounds for mutual stationarity with divergent cofinalities and limited covering

## TL;DR

This paper advances the understanding of mutual stationarity by reducing reliance on covering properties, enabling analysis of sequences with points of countable cofinality, and linking Jf3nsson cardinals to the existence of sharp models with strong cardinals.

## Contribution

It significantly improves previous bounds on mutual stationarity with divergent cofinalities and establishes a connection between Jf3nsson cardinals and the existence of sharp models with strong cardinals.

## Key findings

- Reduced reliance on covering properties in proofs.
- Extended analysis to sequences with infinitely many points of countable cofinality.
- Linked Jf3nsson cardinals to the existence of sharp models with strong cardinals.

## Abstract

We improve previous work on the consistency strength of mutually stationary sequences of sets concentrating on points with divergent cofinality building on previous work by Adolf, Cox and Welch. Specifically, we have greatly reduced our reliance on covering properties in the proof. This will allow us to handle sequences in which sets concentrating on points of countable cofinality appear infinitely often. Furthermore we will show that if $\kappa$ is a J\'onsson cardinal with $\kappa < \aleph_\kappa$ then the sharp for a model with a strong cardinal exists.