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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 ${X\cap \lbrace X\cap \alpha|X\in F}$ 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$do we still get GCH? I am sure if I am asking too much. I just want to see what we already know. PS. What is the right way to pronounce Kurepa? Is it KUrepa (stress on KU), or KuREpa (stress on RE), or KurePA? 2 edited title # GCH+ Kurepa Families 1 # Kurepa Families I have a couple of questions about known theorems for GCH+Kurepa families. Definition first: Let$\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 \alpha|X\in F}$has size$\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?