Aronszajn Free Kurepa Trees

Abstract

We consider a transitive relation on the power set of ${\omega }_1$ and show if there is a maximal element with respect to this relation then there is a Kurepa tree with no Aronszajn subtree. We also show that if there is a maximal subset of ${\omega }_1$, then there are Kurepa trees which are not club isomorphic. These maximal subsets of ${\omega }_1$ exist in many known models that are obtained from the constructible universe without large cardinal assumptions. For instance, whenever ${\alpha }_0\in {\omega }_1$ and $X\subset {\omega }_1$ are such that $ω_1^{\textsc{L}[X \cap α_0]} = ω_1, ω_2^{\textsc{L}[X]} = ω_2$ and $\textsc{V}$ is a semiproper forcing extension of $\textsc{L}[X]$ then $X$ is maximal in $\textsc{V}$.