The generalized Kurepa hypothesis at singular cardinals

We discuss the generalized Kurepa hypothesis $KH_λ$ at singular cardinals $λ$. In particular, we answer questions of Erdös-Hajnal [1] and Todorcevic [6], [7] by showing that $GCH$ does not imply $KH_{\aleph_ω}$ nor the existence of a family $\mathcal{F} \subseteq [\aleph_ω]^{\aleph_0}$ of size $\aleph_{ω+1}$ such that $\mathcal{F} \restriction X$ has size $\aleph_0$ for every $X \subseteq S, |X|=\aleph_0$.

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