The generalized Kurepa hypothesis at singular cardinals

We discuss the generalized Kurepa hypothesis $K{H}_{\lambda }$ at singular cardinals $\lambda$. In particular, we answer questions of Erd"{o}s-Hajnal [1] and Todorcevic [6], [7] by showing that $GCH$ does not imply $K{H}_{{\mathrm{\aleph }}_{\omega }}$ nor the existence of a family $ $\mathcal{F}$ \subseteq [\aleph_\omega]^{\aleph_0}$ofsize$\aleph_{\omega+1}$suchthat$$\mathcal{F}$ \restriction X$hassize$\aleph_0$forevery$X \subseteq S, |X|=\aleph_0$.