Abstrakt
We will review some of the more recent results we have obtained jointly with Sy-David Friedman and Radek Honzik regarding the relationship between the tree property and the continuum function. We will discuss the following key areas: (a).
The possibility of obtaining a strong limit cardinal $\kappa$ with $2^\kappa$ (arbitrarily) large and with the tree property at $\kappa^{++}$. The possibility of having $\kappa = \aleph_\omega$ in the previous result. (b).
The possibility of obtaining the results in (a) from more optimal large-cardinal assumptions (supercompacts vs.\ strong cardinals of low degree). (c). The possibility of obtaining a model where the continuum function below a strong limit $\aleph_\omega$ is as arbitrary as possible with the tree property holding at (some/all) cardinals $\aleph_n$, $1 < n < \omega$.
This work is a part of my PhD project.