Can You Take Komjath's Inaccessible Away?

TL;DR

This paper explores the relationship between Kurepa and Aronszajn trees, demonstrating consistency results under large cardinal assumptions and ZFC, and introduces new techniques involving Todorcevic's rho function.

Contribution

It establishes new consistency results relating Kurepa and Aronszajn trees with minimal large cardinal assumptions and develops a novel lemma about Todorcevic's rho function.

Findings

It is consistent with ZFC that every Kurepa tree contains an Aronszajn subtree under an inaccessible cardinal.

There exists a Kurepa tree with all Kurepa subsets having Aronszajn subtrees without large cardinal assumptions.

A new lemma about Todorcevic's rho function is proved, potentially useful in other contexts.

Abstract

In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains an Aronszajn subtree, if there is an inaccessible cardinal. This is stronger than Komjath's theorem that asserts the same consistency from two inaccessible cardinals. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree $T$ such that if $U \subset T$ is a Kurepa tree with the inherited order from $T$, then $U$ has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: assume $\textrm{MA}_{\omega_2}$ holds and $\omega_2$ is not a Mahlo cardinal in $\textsc{L}$. Then there is a Kurepa tree with the property that every Kurepa subset has an Aronszajn subtree. Our work entails proving a new lemma about Todorcevic's $\rho$ function which might be useful in other contexts.