The generalized Kurepa hypothesis at singular cardinals

Abstract

We discuss the generalized Kurepa hypothesis $K{H}_{\lambda }$ at singular cardinals $\lambda$. In particular, we answer questions of Erdös-Hajnal [1] and Todorcevic [6], [7] by showing that $GCH$ does not imply $K{H}_{{\aleph }_{\omega }}$ nor the existence of a family $\mathcal{F}\subseteq [{\aleph }_{\omega }{]}^{{\aleph }_0}$ of size ${\aleph }_{\omega +1}$ such that $\mathcal{F}\upharpoonright X$ has size ${\aleph }_0$ for every $X\subseteq S,|X|={\aleph }_0$.

arXiv commentIn personal communication, Stevo Todorcevic informed the author that Theorem 3.1 has been obtained by him before; for example can be found on page 231 of his book Walks on ordinals and their characteristics. However our proof is different from him