We discuss the generalized Kurepa hypothesis $${{\mathrm{KH}}}_{\lambda }$$ KH λ at singular cardinals $$\lambda $$ λ . In particular, we answer questions of Erdős–Hajnal (in: Proceedings of the Symposium Pure Mathematics, Part I, University of California, Los Angeles, CA, 1967) and Todorcevic (Trees and linearly ordered sets. Handbook of set-theoretic topology, North-Holland, Amsterdam, pp. 235–293, 1984, Israel J Math 52(1–2): pp. 53–58, 1985) by showing that $${{\mathrm{GCH}}}$$ GCH does not imply $${{\mathrm{KH}}}_{\aleph _\omega }$$ KH ℵ ω nor the existence of a family $$ {\mathcal {F}} \subseteq [\aleph _\omega ]^{\aleph _0}$$ F ⊆ [ ℵ ω ] ℵ 0 of size $$\aleph _{\omega +1}$$ ℵ ω + 1 such that has size $$\aleph _0$$ ℵ 0 for every $$X \subseteq S, |X|=\aleph _0$$ X ⊆ S , | X | = ℵ 0 .