Authors: Lamei Ramandi, Hossein
Todorčević, Stevo 
Affiliations: Mathematics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Can you take Komjath's inaccessible away?
Journal: Annals of Pure and Applied Logic
Volume: 175
Issue: 7
First page: 103452
Issue Date: 1-Jul-2024
Rank: ~M21
ISSN: 0168-0072
DOI: 10.1016/j.apal.2024.103452
Abstract: 
In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the effect of large cardinal assumptions on this comparison. Using 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 in [5], where he proves 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⊂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: If MAω2 holds and ω2 is not a Mahlo cardinal in [Formula presented] 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 ρ function which might be useful in other contexts.
Keywords: Aronszajn trees | Inaccessible cardinals | Kurepa trees | Walks on ordinals
Publisher: Elsevier

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