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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)

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?

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2  
The Wikipedia article shows a pitch accent marker on the u, which means (if the given pronunciation is correct) the first syllable must be stressed. –  Nathan Dec 22 '11 at 23:54
    
Thank you. I always thought it was KuRepa. –  Ioannis Souldatos Dec 23 '11 at 16:02
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1 Answer

up vote 7 down vote accepted

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.

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Philipp, thank you for the answer. Just to clarify: Is this something that has appeared in the literature, or is it something you devised easily yourself? –  Ioannis Souldatos Aug 2 '12 at 18:21
    
Ioannis, I don't know if this specific problem can be found in the literature. I just corrected the last $\kappa^+$ to $\kappa^{++}$. The argument is Silver's proof that $KH(\aleph_1)$ fails after collapsing an inaccessible to $\aleph_2$ adapted to $\kappa^+$. It needs to be checked that the Easton product does not collapse cardinals (compare with Easton forcing in Jech). If $KH(\kappa^{++})$ fails and $\kappa^{++}$ were a successor in $L$, let $x\subseteq \kappa^+$ code its successor, so that $(\kappa^+)^{+L[x]}=\kappa^{++}$. There is a $\kappa^+$-Kurepa tree in $L[x]$ and hence in $V$. –  Philipp Schlicht Aug 6 '12 at 6:26
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The analogous problem for inaccessibles would involve slim Kurepa trees (where each level $\alpha$ has size $\leq|\alpha|$). I just found a relationship between slim Kurepa trees and ineffable cardinals at cantorsattic.info/Ineffable. –  Philipp Schlicht Aug 6 '12 at 6:30
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