arXiv subjects

Šárka Stejskalová

Publications and source records attributed to Šárka Stejskalová.

9 recordsLinked to original sources

arXiv2026

Forcing Over a Free Suslin Tree

We introduce a forcing for adding almost disjoint automorphisms of a normal infinitely splitting $ω_1$-tree $T$ with countable approximations. Assuming that $T$ is a free Suslin tree, this forcing is totally proper, preserves the Suslinness of $T$, and does not add new cofinal branches of $ω_1$-trees existing in intermediate extensions. If $κ$ is an inaccessible cardinal, then the product of the automorphism forcing of length $κ$ with the Lévy collapse of $κ$ to become $ω_2$ forces that there exists an almost Kurepa Suslin tree and there does not exist a Kurepa tree. This model solves open problems due to Bilaniuk, Jin, Shelah, and Moore.

math.LO↗

arXiv2026

Almost Kurepa Suslin trees and destructibility of the Guessing Model Property

Building on recent work of Krueger and the second author, we prove the consistency of the Guessing Model Principle at $ω_2$ together with the existence of an almost Kurepa Suslin tree. In particular, it is consistent that the Guessing Model Principle holds but is destructible by a ccc forcing of size $ω_1$. We also prove the consistency of the existence of a weak Kurepa tree together with the failure of the Kurepa Hypothesis and a certain guessing model principle that, for example, implies the tree property at $ω_2$.

math.LO↗

arXiv2025

A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees

Assuming the negation of Chang's conjecture, there is a c.c.c. forcing which adds a strongly non-saturated Aronszajn tree. Using a Mahlo cardinal, we construct a model in which there exists a strongly non-saturated Aronszajn tree and the negation of the Kurepa hypothesis is c.c.c. indestructible. For any inaccessible cardinal $κ$, there exists a forcing poset which is Y-proper and $κ$-c.c., collapses $κ$ to become $ω_2$, and adds a strongly non-saturated Aronszajn tree. The quotients of this forcing in intermediate extensions are indestructibly Y-proper on a stationary set with respect to any Y-proper forcing extension. As a consequence, we prove from an inaccessible cardinal that the existence of a strongly non-saturated Aronszajn tree is consistent with the non-existence of a weak Kurepa tree. Finally, we prove from a supercompact cardinal that the existence of a strongly non-saturated Aronszajn tree is consistent with two-cardinal tree properties such as the indestructible guessing model principle.

math.LO↗

arXiv2024

Kurepa trees, continuous images, and perfect set properties

Building upon work of Lücke and Schlicht, we study (higher) Kurepa trees through the lens of higher descriptive set theory, focusing in particular on various perfect set properties and representations of sets of branches through trees as continuous images of function spaces. Answering a question of Lücke and Schlicht, we prove that it is consistent with $\mathsf{CH}$ that there exist $ω_2$-Kurepa trees and yet, for every $ω_2$-Kurepa tree $T \subseteq {^{<ω_2}}ω_2$, the set $[T] \subseteq {^{ω_2}}ω_2$ of cofinal branches through $T$ is not a continuous image of ${^{ω_2}}ω_2$. We also produce models indicating that the existence of Kurepa trees is not necessary to produce closed subsets of ${^{ω_1}}ω_1$ failing to satisfy strong perfect set properties, and prove a number of consistency results regarding \emph{full} and \emph{superthin} trees.

math.LO↗

arXiv2024

Club Stationary Reflection and other Combinatorial Principles at $\aleph_{ω+2}$

In this paper we continue the study in [Gilton-Levine-Stejskalova] of compactness and incompactness principles at double successors, focusing here on the case of double successors of singulars of countable cofinality. We obtain models which satisfy the tree property and club stationary reflection at these double successors. Moreover, we can additionally obtain either approachability or its failure. We also show how to obtain our results on $\aleph_{ω+2}$ by incorporating collapses; particularly relevant for these circumstances is a new indestructibility theorem of ours showing that posets satisfying certain linked assumptions preserve club stationary reflection.

math.LO↗

arXiv2023

Strong tree properties, Kurepa trees, and guessing models

We investigate the generalized tree properties and guessing model properties introduced by Weiß and Viale, as well as natural weakenings thereof, studying the relationships among these properties and between these properties and other prominent combinatorial principles. We introduce a weakening of Viale and Weiß's Guessing Model Property, which we call the Almost Guessing Property, and prove that it provides an alternate formulation of the slender tree property in the same way that the Guessing Model Property provides and alternate formulation of the ineffable slender tree property. We show that instances of the Almost Guessing Property have sufficient strength to imply, for example, failures of square or the nonexistence of weak Kurepa trees. We show that these instances of the Almsot Guessing Property hold in the Mitchell model starting from a strongly compact cardinal and prove a number of other consistency results showing that certain implications between the principles under consideration are in general not reversible. In the process, we provide a new answer to a question of Viale by constructing a model in which, for all regular $θ\geq ω_2$, there are stationarily many $ω_2$-guessing models $M \in \mathscr{P}_{ω_2} H(θ)$ that are not $ω_1$-guessing models.

math.LO↗

arXiv2023

Guessing models, trees, and cardinal arithmetic

Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the Guessing Model Property and cardinal arithmetic, broadly construed. We improve upon results of Viale and Krueger by proving that a weakening of the Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that, though the Guessing Model Property is known not to put an upper bound on the size of the continuum, it does imply that $2^{ω_1}$ is as small as possible relative to the value of $2^ω$. Building on work of Laver, we prove that, in the extension of any model of $\mathsf{PFA}$ by a measure algebra, every tree of height and size $ω_1$ is B-special (a generalization of specialness introduced by Baumgartner that can also hold of trees with uncountable branches). Finally, we investigate the impact of forcing axioms for Suslin and almost Suslin trees on guessing model properties. In particular, we prove thatif $S$ is a Suslin tree, then the axioms $\mathsf{PFA}(S)$ and $\mathsf{PFA}(S)[S]$ imply the Guessing Model Property and the Indestructible Guessing Model Property, respectively, and, if $T^*$ is an almost Suslin Aronszajn tree, then the axiom $\mathsf{PFA}(T^*)$ implies the Indestructible Guessing Model Property. This answers a number of questions of Cox and Krueger.

math.LO↗

arXiv2022

Indestructibility of some compactness principles over models of PFA

We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $ω$ will not add an $ω_2$-Aronszajn tree or a weak $ω_1$-Kurepa tree, and moreover no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree (a tree of height and size $ω_1$ with at least $ω_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add $ω_2$-Aronszajn or $ω_1$-Kurepa trees. We actually prove more: We show that a consequence of $\mathsf{PFA}$, namely the guessing model principle, $\mathsf{GMP}$, which is equivalent to the ineffable slender tree property, $\mathsf{ISP}$, is preserved by adding any number of Cohen subsets of $ω$. And moreover, $\mathsf{GMP}$ implies that no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree. For more generality, we study the principle $\mathsf{GMP}$ at an arbitrary regular cardinal $κ= κ^{<κ}$ (we denote this principle $\mathsf{GMP}_{κ^{++}}$), and as an application we show that there is a model in which there are no weak $\aleph_{ω+1}$-Kurepa trees and no $\aleph_{ω+2}$-Aronszajn trees.

math.LO↗

arXiv2021

Trees and stationary reflection at double successors of regular cardinals

We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $κ$, updating some classical constructions in the process. This includes models of $\mathsf{CSR}(κ^{++})\wedge \mathsf{TP}(κ^{++})$ (both with and without $\mathsf{AP}(κ^{++})$) and models of the conjunctions $\mathsf{SR}(κ^{++}) \wedge \mathsf{wTP}(κ^{++}) \wedge \mathsf{AP}(κ^{++})$ and $\neg \mathsf{AP}(κ^{++}) \wedge \mathsf{SR}(κ^{++})$ (the latter was originally obtained in joint work by Krueger and the first author \cite{GilKru:8fold}, and is here given using different methods). Analogs of these results with the failure of $\mathsf{SH}(κ^{++})$ are given as well. Finally, we obtain all of our results with an arbitrarily large $2^κ$, applying recent joint work by Honzik and the third author.

math.LO↗