# Global Chang's Conjecture and singular cardinals

## TL;DR

This paper explores the potential for global versions of Chang's Conjecture involving singular cardinals, establishing ZFC limitations and consistency results relative to large cardinals.

## Contribution

It demonstrates ZFC limitations and shows that Chang's Conjecture can be consistently extended between all pairs of limit cardinals below .

## Key findings

- ZFC imposes limitations on global Chang's Conjecture with singular cardinals
- Consistency of Chang's Conjecture between all limit cardinals below  established relative to large cardinals
- Results advance understanding of the interplay between singular cardinals and Chang's Conjecture

## Abstract

We investigate the possibilities of global versions of Chang's Conjecture that involve singular cardinals. We show some $\mathrm{ZFC}$ limitations on such principles, and prove relative to large cardinals that Chang's Conjecture can consistently hold between all pairs of limit cardinals below $\aleph_{\omega^\omega}$.