# Towers and clubs

## TL;DR

This paper investigates club principles and the nonsaturation of the nonstationary ideal on regular uncountable cardinals, aiming to improve existing results by constructing towers of various lengths to demonstrate nonsaturation.

## Contribution

It extends previous work by considering non-normal ideals and constructing towers of length possibly greater than ^+ to witness nonsaturation.

## Key findings

- Established new conditions for nonsaturation of ideals.
- Constructed towers of length exceeding ^+ in certain models.
- Improved bounds related to club principles and nonstationary ideals.

## Abstract

We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a (non necessarily normal) ideal $J$ extending the nonstationary ideal on a regular uncountable (non necessarily successor) cardinal $\kappa$, our goal being to witness the nonsaturation of $J$ by the existence of towers (of length possibly greater than $\kappa^+$).