I have a couple of questions about known theorems for GCH+Kurepa families.
Definition first: Let $\kappa$ be a infinite cardinal. A $\kappa^+$ Kurepa family is a family $F$ of subsets of $\kappa^+$ such that $F$ has size $>\kappa^+$ and for every $a<\kappa^+$, the set ${X\cap \lbrace X\cap \alpha|X\in F}$ F\rbrace$ has size $\le\kappa$.
(The definition can be given in terms of tress too).
$KH(\kappa^+)$ is the statement that a $\kappa^+$ Kurepa family exists.
Please correct me, if I am mistaken, but we know that $KH(\kappa^+)$ holds for all infinite $\kappa$ in $L$ (the constructible universe). Also, if $\lambda$ is an inaccessible cardinal and we collapse $\lambda$ to $\aleph_2$, then in the generic extension $KH(\aleph_1)$ fails. (Look also this http://mathoverflow.net/questions/68533)
So, my questions are:
1) Do we know any models where GCH holds and $KH(\kappa^+)$ fails for all $\kappa$?
2) If this is not the case, can we at least have GCH+ the failure of $KH(\aleph_{\alpha+1})$, for all $\alpha$ countable ordinals?
3) If (2) is not known either, then fix some $\alpha$ countable ordinal $>0$. Can we have GCH+ the failure of $KH(\aleph_{\alpha+1})$?
4) If the ground model satisfies GCH, after we collapse an inaccessible cardinal to $\aleph_2$ do we still get GCH?
I am sure if I am asking too much. I just want to see what we already know.
PS. What is the right way to pronounce Kurepa? Is it KUrepa (stress on KU), or KuREpa (stress on RE), or KurePA?
