Can a small forcing create Kurepa trees (Q1356977)

The aim of the paper is to examine the possibility of creating Kurepa trees in generic extensions of a ground model satisfying CH by an \(\omega_1\)-preserving forcing notion of size at most \(\omega_1\). In the literature there are several known ways of generic constructions of a Kurepa tree (D. H. Stewart, J. H. Silver, B. Veličković,\dots) but the used forcing notions have size at least \(\omega_2\) to be able to guarantee that the generic tree has at least \(\omega_2\) branches. In the Lévy model forcing with \(\text{Coll}(\omega,\omega_1)\) produces a Kurepa tree so to avoid this triviality \(\omega_1\)-preserving is required. In the first part of the paper the results show some evidence that in the Lévy model \(V\) it is extremely hard to find such a forcing notion \(P\), if it ever exists. If \(P\) does not add countable sequences of ordinals, or if \(S\subseteq\omega_1\) is stationary and \(P\) is an \((S,\omega)\)-proper forcing notion (which includes \(\omega\)-proper and Axiom A forcing notions) of size \(\leq\omega_1\), then there are no Kurepa trees in \(V^P\). The Baumgartner and Taylor forcing notion for adding a club subset of \(\omega_1\) by finite conditions has size \(\omega_1\) and is proper but not \((S,\omega)\)-proper. The authors find a general forcing property which this forcing has and prove that all such forcing notions do not add Kurepa trees over the Lévy model. In the second part, assuming that \(\kappa\) is a strongly inaccessible cardinal, the authors define an \(\omega_1\)-closed \(\kappa\)-c.c.\ forcing notion collapsing \(\kappa\) to \(\omega_2\) which produces a model of CH with no Kurepa trees but with an \(\omega\)-distributive Aronszajn tree \(T\) such that the forcing with \(T\) produces a Kurepa tree.