Aronszajn Free Kurepa Trees

TL;DR

This paper explores the existence of special Kurepa trees under certain set-theoretic assumptions, showing how maximal subsets of  lead to trees with unique properties in various models.

Contribution

It establishes the connection between maximal subsets of  and the existence of Kurepa trees with specific structural features, under models derived from the constructible universe.

Findings

Existence of Kurepa trees with no Aronszajn subtree under certain maximality conditions.

Maximal subsets of  imply the existence of non-club-isomorphic Kurepa trees.

Such maximal subsets are present in many models obtained via semiproper forcing extensions.

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 $\omega_1^{\textsc{L}[X \cap \alpha_0]} = \omega_1, \omega_2^{\textsc{L}[X]} = \omega_2$ and $\textsc{V}$ is a semiproper forcing extension of $\textsc{L}[X]$ then $X$ is maximal in $\textsc{V}$.