# Ramsey cardinals and the continuum function

## Victoria Gitman

### The City University of New York

In his famous theorem, Easton used the Easton Product forcing to show that if $V\models{\rm GCH}$ and $F$ is any weakly increasing function on the regular cardinals such that $\text{cf}(F(\alpha))>\alpha$, then there is a cofinality preserving forcing extension in which $F$ is realized as the continuum function. The investigation then shifted to identifying which continuum patterns are compatible with large cardinals. It is not difficult to see that large cardinals affect the behavior of the continuum function. Obviously, if $\kappa$ is inaccessible, then by definition, the continuum function must have a closure at $\kappa$. Some other large cardinal influences are much more subtle. Easton’s original forcing does not work well in the presence of large cardinals; it, for instance, destroy weak compactness over $L$. So set theorists have had to develop some general and other very specific forcing techniques to address the behavior of the continuum function for a given large cardinal. In this talk, we will show that if $V\models{\rm GCH}$, $\kappa$ is Ramsey, and $F$ is any weakly increasing class function on the regular cardinals with a closure point at $\kappa$ such that $\text{cf}(F(\alpha))>\alpha$, then there is a cofinality preserving forcing extension in which $\kappa$ remains Ramsey and $F$ is realized as the continuum function. This is joint work with Brent Cody.

An extended abstract can be found here.

Victoria Gitman received her Ph.D. in 2007 from the CUNY Graduate Center, as a student of Joel Hamkins, and is presently a visiting scholar at the CUNY Graduate Center. Her research is in Mathematical Logic, in particular in the areas of Set Theory and Models of Peano Arithmetic.