Higher Chang Conjecture

In this talk I will present some results regarding the consistency strength of Higher variants of Chang's Conjecture. 

I will start with the classical result by Silver of Chang's Conjecture from $\omega_1$-Erdos cardinal.  

Then, I will give an upper bound for the consistency strength of  $(\aleph_{\omega+1}, \aleph_{\omega}) -->>(\aleph_1, \aleph_0)$

and $(\aleph_4, \aleph_3) -->> (\aleph_2, \aleph_1)$ (joint with Eskew)

from supercompactness assumptions. 

If time permits, I will describe the strategy for obtaining a global result:

(\kappa^+,\kappa) -->> (\mu^+, \mu)

for all regular $\kappa$, and $\mu < \kappa$, and talk about the barriers that we face when trying to extend this result.

 

Link to recording.