The generalized Kurepa hypothesis at singular cardinals

TL;DR

This paper investigates the generalized Kurepa hypothesis at singular cardinals, demonstrating that GCH does not imply certain combinatorial properties at $eth_ ext{omega}$, thus answering longstanding open questions.

Contribution

It shows that GCH does not imply the generalized Kurepa hypothesis at $eth_ ext{omega}$, resolving questions posed by Erdős-Hajnal and Todorcevic.

Findings

GCH does not imply $KH_{eth_ ext{omega}}$

No family of size $eth_ ext{omega+1}$ with countable restrictions exists at $eth_ ext{omega}$

Answers to Erdős-Hajnal and Todorcevic questions

Abstract

We discuss the generalized Kurepa hypothesis $KH_{\lambda}$ 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 $KH_{\aleph_\omega}$ nor the existence of a family $ \mathcal{F} \subseteq [\aleph_\omega]^{\aleph_0}$ of size $\aleph_{\omega+1}$ such that $\mathcal{F} \restriction X$ has size $\aleph_0$ for every $X \subseteq S, |X|=\aleph_0$.