Hello,


I want to ask if anyone can tells us what is known (consistently) about $Ded(\kappa)$, $\kappa$ an infinite cardinal. 


**Definition** If there is a dense linear order w/o endpoints of size $\lambda$ with a dense subset of size $\kappa$ then we write $D(\kappa,\lambda)$. Then $Ded(\kappa)=\sup\_\lambda \{D(\kappa,\lambda)\}$.


**Known theorems:** (1) $Ded(\kappa)\le 2^\kappa$ and under GCH $Ded(\kappa)$ is always equal to $2^\kappa$. 
(2) If $\mu$ is the least cardinal such that $\kappa^\mu>\kappa$, then $D(\kappa,\kappa^\mu)$ holds, which implies in particular that $Ded(\kappa)\ge \kappa^\mu$. 


**Questions** (1) Can we prove that $Ded(\kappa)< Ded(\kappa)^\omega$ is consistent?
(2) If $\mu$ a cardinal between $\omega$ and $\kappa$, can we prove that $Ded(\kappa)=\kappa^\mu$ is consistent?


Note 1: Following Keisler $Ded(\kappa)$, $Ded(\kappa)^\omega$ are two of the six possible "stability functions", the other four being $\kappa$, $\kappa+2^\omega$, $\kappa^\omega$ and $2^\kappa$. Stability functions give us the number of types of a theory $T$ over models of power $\kappa$. For more on this consult *The Stability Function of a Theory* by 
H. Jerome Keisler, The Journal of Symbolic Logic, Vol. 43, No. 3 (Sep., 1978), pp. 481-486 


Note 2: There is a similar question posted on MathOverflow ([Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?](https://mathoverflow.net/questions/48231)) that asks for the consistency of $Ded(\kappa)<2^\kappa$ (answer is positive)



Is $Ded(\kappa)<Ded(\kappa)^\omega$ consistent?
I just found the following paper on arXiv: "On non-forking spectra" by Artem Chernikov, Itay Kaplan and Saharon Shelah ( <http://arxiv.org/abs/1205.3101> ). They claim that it is consistent that $Ded(\kappa)< Ded(\kappa)^\omega$, therefore answering this question positively. 


