Aronszajn Free Kurepa Trees
Hossein Lamei Ramandi, Stevo Todorcevic

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 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 , then there are Kurepa trees which are not club isomorphic. These maximal subsets of exist in many known models that are obtained from the constructible universe without large cardinal assumptions. For instance, whenever and are such that and is a semiproper forcing extension of then is maximal in .
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Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms · Economic theories and models
