Global Chang’s Conjecture and singular cardinals

@article{Eskew2021GlobalCC,
  title={Global Chang’s Conjecture and singular cardinals},
  author={Monroe Eskew and Yair Hayut},
  journal={European Journal of Mathematics},
  year={2021},
  volume={7},
  pages={435 - 463}
}
We investigate the possibilities of global versions of Chang’s Conjecture that involve singular cardinals. We show some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{ZFC} $$\end{document}ZFC limitations on such principles and prove relative to large cardinals that Chang’s Conjecture can consistently hold… 
1 Citations
Compactness versus hugeness at successor cardinals
If $\kappa$ is regular and $2^{<\kappa}\leq\kappa^+$, then the existence of a weakly presaturated ideal on $\kappa^+$ implies $\square^*_\kappa$. This partially answers a question of Foreman and

References

SHOWING 1-10 OF 37 REFERENCES
The large cardinals between supercompact and almost-huge
TLDR
The hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal is analyzed, and some results relating high-jump cardinals to forcing are proved, as well as analyzing Laver functions for super-high- jump cardinals.
Reflecting stationary sets and successors of singular cardinals
TLDR
It is shown that supercompactness (and even the failure of PT) implies the existence of non-reflecting stationary sets, and that under suitable assumptions it is consistent that REF and there is a κ which is κ+n-supercompact.
Squares, scales and stationary Reflection
TLDR
Interactions between these three theories in the context of singular cardinals are considered, focusing on the various implications between square and scales (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.
When does almost free imply free? (For groups, transversals, etc.)
We show that the construction of an almost free nonfree Abelian group can be pushed from a regular cardinal /C to ~IC+I. Hence there are unboundedly many almost free nonfree Abelian groups below the
A Very Weak Square Principle
TLDR
A very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L , which is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals.
Prikry-Type Forcings
One of the central topics of set theory since Cantor has been the study of the power function κ→2 κ . The basic problem is to determine all the possible values of 2 κ for a cardinal κ. Paul Cohen
Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II
We prove, assuming the existence of a huge cardinal, the consistency of fully non-regular ultrafilters on the successor of any regular cardinal. We also construct ultrafilters with ultraproducts of
The tree property on a countable segment of successors of singular cardinals
Starting from the existence of many supercompact cardinals, we construct a model of ZFC in which the tree property holds at a countable segment of successor of singular cardinals.
...
...