Easton's Theorem for Ramsey and Strongly Ramsey cardinals

TL;DR

This paper proves that under GCH, the continuum function can be controlled via forcing while preserving the Ramsey or strongly Ramsey nature of a cardinal, extending Easton's theorem to these large cardinals.

Contribution

It extends Easton's theorem to Ramsey and strongly Ramsey cardinals, showing the continuum function can be freely prescribed without destroying their large cardinal properties.

Findings

The continuum function can be set to any Easton-allowed pattern while preserving Ramsey cardinals.

The continuum function can be set to any Easton-allowed pattern while preserving strongly Ramsey cardinals.

The results depend on GCH and involve cofinality-preserving forcing extensions.

Abstract

We show that, assuming GCH, if $\kappa$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $\kappa$ and obeying the constraints of Easton's theorem, namely, $F(\alpha)\leq F(\beta)$ for $\alpha\leq\beta$ and $\alpha<\cf(F(\alpha))$, then there is a cofinality preserving forcing extension in which $\kappa$ remains Ramsey or strongly Ramsey respectively and $2^\delta=F(\delta)$ for every regular cardinal $\delta$.