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 $\lbrace X\cap \alpha|X\in F\rbrace$ 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 [On the independence of the Kurepa Hypothesis](https://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?



GCH+ Kurepa Families
Hi Ioannis! I guess you might know the answer by now; if we suppose for 1 that there is a class of inaccessible cardinals in the ground model and force with an Easton support product of Lévy collapses between the inaccessibles, we obtain a model of GCH where $KH(\kappa^+)$ fails for all $\kappa$; the argument is the same as for $\omega\_1$. Also the failure of $KH(\kappa^+)$ implies that $\kappa^{++}$ is inaccessible in $L$, so we need the inaccessibles. 


