# Local saturation and square everywhere

## TL;DR

This paper demonstrates, assuming large cardinal axioms, that for all infinite cardinals, the square principle holds and there exists a stationary set with a highly saturated non-stationary ideal, linking combinatorial principles with saturation properties.

## Contribution

It establishes the consistency of square principles and saturation of non-stationary ideals simultaneously across all infinite cardinals, under large cardinal assumptions.

## Key findings

- Square principles hold for all infinite cardinals.
- Existence of a stationary set with a highly saturated non-stationary ideal.
- Consistency results relative to large cardinals.

## Abstract

We show that it is consistent relative to a huge cardinal that for all infinite cardinals $\kappa$, $\square_\kappa$ holds and there is a stationary $S \subseteq \kappa^+$ such that $\mathrm{NS}_{\kappa^+} \restriction S$ is $\kappa^{++}$-saturated.