At the present there are at least three published paper concerning the global behavior of the power function:

1-Foreman-Woodin, The generalized continuum hypothesis can fail everywhere.

2- Cummings, A model in which GCH holds at successors but fails at limits.

3-Merimovich, A power function with a fixed finite gap everywhere.

The first paper uses a supercompact cardinal with infinitely many inaccessibles above it, while the last two papers use strong cardinals. Let me mention that:

(*) In all of these models cofinalities are changed and in the last two models cardinals are also collapsed.

(**) All of these models are obtained in two steps: At the first step a reverse Easton iteration is done which blows up the power of some cardinals below $\kappa$ ($\kappa$ is the large cardinal which we are using to exist in the ground model) and in the extension some guiding generics are constructed for the use of second step. In the second step a variant of Radin forcing (usually with interleaved collapsed,...) is used. Note that changing cofinalities and collapsing cardinals are presented in this step. At the end $\kappa$ remains inaccessible and below $\kappa$ we have the behavior of the function as we requested.

I would like to mention two recent results of Sy Friedman and I:

**Theorem 1.** Assuming the existence of a $\kappa+3-$strong cardinal $\kappa,$ there exists a pair $(W,V)$ of models of $ZFC$ such that:

1- $W$ and $V$ ahve the same cardinals,

2-GCH holds in $W$,

3-$GCH$ fails everywhere in $V$,

4-$V=W[R]$ for some real $R$.

The above theorem says that it is possible to kill the GCH everywhere just by adding a single real. It answers an open question of Shelah-Woodin "Forcing the failure of CH by adding a real" (I like the above result a lot).

**Theorem 2.** Assuming the existence of a $\kappa+4-$strong cardinal $\kappa$ it is consistent to have a pair $(W,V)$ of models of $ZFC$ such that:

1-$W$ and $V$ have the same cofinalities,

2-GCH holds in $W$,

3-$V\models \forall \lambda, 2^{\lambda}=\lambda^{+3}.$

Thus it is possible to kill the GCH everywhere by a cofinality preserving forcing.

I also should mention that Moti Gitik and Carmi Merimovich are doing a project in which they are planning to obatain the global failure of GCH from the optimal hypotheses. If I remeber their result correctly it says something like this:

**Theorem.** The following are equiconsistent:

1-For any $\alpha,$ there are stationary many cardinals $\kappa$ with $O(\kappa)=\kappa^{++}+\alpha,$

2-GCH fails everywhere,

3-$\forall \lambda, 2^{\lambda}=\lambda^{++}.$

Gitik told me that their proof uses the Radin forcing and they need to embed many models into each other. My guess is that the paper " Gitik, Moti; Merimovich, Carmi Power function on stationary classes" is related to their work.

A Power Function with a Fixed Finite Gap Everywhere, The Journal of Symbolic Logic,72 (2), (2007), 361-417. Merimovich uses extender based Radin forcing, his argument can give $\forall\lambda\,(2^\lambda=\lambda^{+n})$ for any fixed $n$, $1<n<\omega$, though he presents the details for $n=3$. $\endgroup$1more comment