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We discuss the generalized Kurepa hypothesis \({{\mathrm{KH}}}_{\lambda }\) at singular cardinals \(\lambda \). In particular, we answer questions of Erdï¿½?sï¿½??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. 235ï¿½??293, 1984, Israel J Math 52(1ï¿½??2): pp. 53ï¿½??58, 1985) by showing that \({{\mathrm{GCH}}}\) does not imply \({{\mathrm{KH}}}_{\aleph _\omega }\) nor the existence of a family \( {\mathcal {F}} \subseteq [\aleph _\omega ]^{\aleph _0}\) of size \(\aleph _{\omega +1}\) such that Open image in new window has size \(\aleph _0\) for every \(X \subseteq S, |X|=\aleph _0\).
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