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John Krueger

Publications and source records attributed to John Krueger.

At least 19 recordsLinked to original sources
arXiv2026

Countryman Lines and the Continuum Hypothesis

We explore the prospect of basis-like results for the class of Aronszajn lines which are consistent with the Continuum Hypothesis (CH). In particular, we prove that each of the following statements is consistent with CH: Any two Countryman lines contain isomorphic or anti-isomorphic uncountable suborders; for any coherent Aronszajan tree $T \subseteq {}^{< \omega_1} \omega$, the filter $\mathcal{U}(T)$ is an ultrafilter. The first result confirms a conjecture of Shelah in the context of CH which was previously shown to follow from the Proper Forcing Axiom. On the other hand, the weak diamond principle $2^\omega < 2^{\omega_1}$ implies that there does not exist a two element basis for the Countryman lines.

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arXiv2026

Combinatorial Properties Related to the Higher Baumgartner's Axiom

We isolate two combinatorial properties, each expressible by a $\Pi_2$-sentence over the structure $(H(\omega_3),\in,\omega_1,\omega_2,\text{NS}_{\omega_2})$, such that each property is consistent with CH, and their conjunction together with $2^\omega \le \omega_2$ and $2^{\omega_1} = 2^{\omega_2} = \omega_3$ implies the existence of a c.c.c. forcing which forces the higher Baumgartner's axiom.

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arXiv2025

A virtual five element basis for the uncountable linear orders

We prove that for every Aronzsajn line A and every Countryman line C, there is a proper forcing extension in which A contains an isomorphic copy of either C or its converse C*. As a corollary, we obtain answers to several related questions asked by the second author in the literature: if there is an inaccessible cardinal, then there is a proper forcing extension in which the uncountable linear orders have a five element basis; BPFA implies the existence of a five element basis for the uncountable linear orders; BPFA is equiconsistent with the conjunction of BPFA and Aronszajn tree saturation. These results are derived from new preservation results concerning subtrees of Aronszajn trees, proper forcings, and countable support iterations, generalizing work of Miyamoto, Abraham, and Shelah.

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arXiv2025

A Strong Kurepa Tree

We prove that it is consistent that there exists a Kurepa tree $T$ such that ${}^{\omega_1}2$ is a continuous image of the topological space $[T]$ consisting of all cofinal branches of $T$ with respect to the cone topologies. This result solves an open problem due to Bergfalk, Chodounsk\'{y}, Guzm\'{a}n, and Hru\v{s}\'{a}k. We also prove that any Kurepa tree with the above property contains an Aronszajn subtree.

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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 $\kappa$, there exists a forcing poset which is Y-proper and $\kappa$-c.c., collapses $\kappa$ to become $\omega_2$, 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.

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arXiv2024

Some Results on Finitely Splitting Subtrees of Aronszajn Trees

For any $2 \le n < \omega$, we introduce a forcing poset using generalized promises which adds a normal $n$-splitting subtree to a $(\ge \! n)$-splitting normal Aronszajn tree. Using this forcing poset, we prove several consistency results concerning finitely splitting subtrees of Aronszajn trees. For example, it is consistent that there exists an infinitely splitting Suslin tree whose topological square is not Lindel\"{o}f, which solves an open problem due to Marun. For any $2 < n < \omega$, it is consistent that every $(\ge \! n)$-splitting normal Aronszajn tree contains a normal $n$-splitting subtree, but there exists a normal infinitely splitting Aronszajn tree which contains no $(< \! n)$-splitting subtree. To show the latter consistency result, we prove a forcing iteration preservation theorem related to not adding new small-splitting subtrees of Aronszajn trees.

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arXiv2024

An almost Kurepa Suslin tree with strongly non-saturated square

For uncountable downwards closed subtrees $U$ and $W$ of an $\omega_1$-tree $T$, we say that $U$ and $W$ are strongly almost disjoint if their intersection is a finite union of countable chains. The tree $T$ is strongly non-saturated if there exists a strongly almost disjoint family of $\omega_2$-many uncountable downwards closed subtrees of $T$. In this article we construct a Knaster forcing which adds a Suslin tree together with a family of $\omega_2$-many strongly almost disjoint automorphisms of it (and thus the square of the Suslin tree is strongly non-saturated). To achieve this goal, we introduce a new idea called $\rho$-separation, which is an adaptation to the finite context of the notion of separation which was recently introduced by Stejskalov\'{a} and the first author for the purpose of adding automorphisms of a tree with a forcing with countable conditions.

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arXiv2024

Forcing Over a Free Suslin Tree

We introduce a forcing for adding almost disjoint automorphisms of a normal infinitely splitting $\omega_1$-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 $\omega_1$-trees existing in intermediate extensions. If $\kappa$ is an inaccessible cardinal, then the product of the automorphism forcing of length $\kappa$ with the L\'{e}vy collapse of $\kappa$ to become $\omega_2$ 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.

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arXiv2023

A Rigid Kurepa Tree From a Free Suslin Tree

We analyze a countable support product of a free Suslin tree which turns it into a highly rigid Kurepa tree with no Aronszajn subtree. In the process, we introduce a new rigidity property for trees, which says roughly speaking that any non-trivial strictly increasing function from a section of the tree into itself maps into a cofinal branch.

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arXiv2022

A Large Pairwise Far Family of Aronszajn Trees

We construct a large family of normal $κ$-complete $\mathbb{R}_κ$-embeddable non-special $κ^+$-Aronszajn trees which have no club isomorphic subtrees using an instance of the proxy principle of Brodsky-Rinot.

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arXiv2022

Suslin tree preservation and club isomorphisms

We construct a model of set theory in which there exists a Suslin tree and satisfies that any two normal Aronszajn trees, neither of which contains a Suslin subtree, are club isomorphic. We also show that if $S$ is a free normal Suslin tree, then for any positive integer $n$ there is a c.c.c. forcing extension in which $S$ is $n$-free but all of its derived trees of dimension greater than $n$ are special.

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arXiv2020

A forcing axiom for a non-special Aronszajn tree

Suppose that $T^*$ is an $ω_1$-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA($T^*$) for proper forcings which preserve these properties of $T^*$. We prove that PFA($T^*$) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than $ω_1$, and the $P$-ideal dichotomy. On the other hand, PFA($T^*$) implies some of the consequences of diamond principles, such as the existence of Knaster forcings which are not stationarily Knaster.

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arXiv2020

Entangledness in Suslin lines and trees

We introduce the idea of a weakly entangled linear order, and show that it is consistent for a Suslin line to be weakly entangled. We generalize the notion of entangled linear orders to $ω_1$-trees, and prove that an $ω_1$-tree is entangled iff it is free. We force the existence of a Suslin tree which is $n$-entangled, but all of whose derived trees of dimension $n+1$ are special, for any positive $n < ω$.

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arXiv2019

Parametrized Measuring and Club Guessing

We introduce Strong Measuring, a maximal strengthening of J. T. Moore's Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of $ω_1$ is measured by some club subset of $ω_1$. The consistency of Strong Measuring with the negation of CH is shown, solving an open problem from about parametrized measuring principles. Specifically, we prove that Strong Measuring follows from MRP together with Martin's Axiom for $σ$-centered forcings, as well as from BPFA. We also consider strong versions of Measuring in the absence of the Axiom of Choice.

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arXiv2019

A note on the eightfold way

Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an $ω_2$-Aronszajn tree, the $ω_1$-approachability property fails, and every stationary subset of $ω_2 \cap \mathrm{cof}(ω)$ reflects.

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arXiv2019

Guessing models imply the singular cardinal hypothesis

In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $κ\ge ω_2$, $\textsf{ISP}(κ)$ implies that $\textsf{SCH}$ holds above $κ$, and (3) forcing posets which have the $ω_1$-approximation property also have the countable covering property. These results solve open problems of Viale and Hachtman-Sinapova.

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arXiv2019

The Harrington-Shelah Model with Large Continuum

We prove from the existence of a Mahlo cardinal the consistency of the statement that $2^ω= ω_3$ holds and every stationary subset of $ω_2 \cap \mathrm{cof}(ω)$ reflects to an ordinal less than $ω_2$ with cofinality $ω_1$.

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