Ralf Schindler
Suppose that there's no transitive model of ZFC + there's a strong cardinal, and let K denote the core model. It is shown that if δhas the tree property then δ^{+K} = δ^+ and δis weakly compact in K.
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In this paper we review Shelah's strong covering property and its applications. We also extend some of the results of Shelah and Woodin on the failure of $CH$ by adding a real.
An optimal extension of the Jensen covering lemma, within the limits imposed by Prikry forcing, is proved. If L[E] is an "iterable" weasel with no measurable cardinals, then either L[E] has "indiscernibles", or every uncountable set of ordinals is contained in a set in L[E] of the same cardinality. (The terms "iterable" and "indiscernibles" are made precise in the paper.) Most importantly, there is no hypothesis explicitly limiting the large cardinals which are consistent in L[E].
We construct a model in which the tree property holds in $\aleph_{ω+ 1}$ and it is destructible under $\text{Col}(ω, ω_1)$. On the other hand we discuss some cases in which the tree property is indestructible under small or closed forcings.
We prove that under certain set-theoretic assumptions every productively Lindelöf space has the Hurewicz covering property, thus improving upon some earlier results of Aurichi and Tall.
Assuming the existence of a proper class of supercompact cardinals, we force that for every regular cardinal $κ$, there are $κ^+$-Aronszajn trees and all such trees are special.
Starting with infinitely many supercompact cardinals, we show that the tree property at every cardinal $\aleph_n$, $1 < n <ω$, is consistent with an arbitrary continuum function below $\aleph_ω$ which satisfies $2^{\aleph_n} > \aleph_{n+1}$, $n<ω$. Thus the tree property has no provable effect on the continuum function below $\aleph_ω$ except for the restriction that the tree property at $κ^{++}$ implies $2^κ>κ^+$ for every infinite $κ$.
Starting from the existence of many supercompact cardinals, we construct a model of ZFC in which the tree property holds at a countable segment of successor of singular cardinals.
We show that it is consistent, relative to $ω$ many supercompact cardinals, that the super tree property holds at $\aleph_n$ for all $2 \leq n < ω$ but there are weak square and a very good scale at $\aleph_ω$.
Starting from a stationary set of supercompact cardinals we find a generic extension in which the tree property holds at every regular cardinal between $\aleph_2$ and $\aleph_{ω^2}$.
It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to indescribables but are much weaker in consistency strength. Also the consistency strength of definable tree property for successor of a singular cardinal is reduced to the existence of a supercompact cardinal and a measurable above it.
Assuming the existence of a strong cardinal $κ$ and a measurable cardinal above it, we force a generic extension in which $κ$ is a singular strong limit cardinal of any prescribed cofinality, and such that the tree property holds at $κ^{++}$.
Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.
Assume $λ$ is a singular limit of $η$ supercompact cardinals, where $η\leq λ$ is a limit ordinal. We present two forcing methods for making $λ^+$ the successor of the limit of the first $η$ measurable cardinals while the tree property holding at $λ^+.$ The first method is then used to get, from the same assumptions, tree property at $\aleph_{η^2+1}$ with the failure of $SCH$ at $\aleph_{η^2}$. This extends results of Neeman and Sinapova. The second method is also used to get tree property at successor of an arbitrary singular cardinal, which extends some results of Magidor-Shelah, Neeman and S…
The tree property on long intervals of regular cardinals
本文围绕 Foreman–Magidor 关于首个不可数基数以上所有正则基数均具有树性质的问题，证明树性质可在与强极限基数相交的正则基数区间上成立。
Assuming the existence of a strong cardinal $κ$, a weakly compact cardinal $λ$ above it and $γ> λ,$ we force a generic extension in which $κ$ is a singular strong limit cardinal of any given cofinality $δ$, $2^κ\geq γ$ and such that the tree property holds at $κ^{++}$.
Definable tree property for uncountable regular cardinals
该研究以一个超紧基数和一个更大的可测基数为基础，构造出满足所有不可数正则基数都具有可定义树性质的ZFC模型，回应了相关文献中的问题。
We establish a surprising connection between Menger's classical covering property and Blass's modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger's property.
The Strong and Super Tree Property at Successors of Singular Cardinals
该研究聚焦奇异基数后继的强树性质与ITP（超树性质），构造通用力迫类，推广Neeman的结果，证明多种基数组合下两类性质同时成立的一致性，推进了同时在多个相继基数上获得两类树性质的问题研究。
We force the existence of a non-trivial $κ$-complete ultrafilter over $κ$ which fails to satisfy the Galvin property. This answers a question asked by the first author and Moti Gitik.
Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $δ$, $δ$ is strong limit, $2^δ=δ^{+3}$ and the tree property at $δ^{++}$ holds. This answers a question of Friedman, Honzik and Stejskalova [8]. We also produce, relative to the existence of a strong cardinal and two measurable cardinals above it, a model of $ZFC$ in which the tree property holds at all regular even cardinals. The result answers questions of Friedman-Halilovic [5] and Friedman-Honzik [6].
Slender Trees and the Approximation Property
本文推广了Unger的结果，提出了判断力迫具有${<}\,δ$-近似性质的简单准则，并据此构造Mitchell Forcing变体以回答Mohammadpour的问题。
We show that for any regular cardinal $κ$, $\square_{κ, 2}$ is consistent with "all $κ^+$-Aronszajn trees are special." By a result of Shelah and Stanley this is optimal in the sense that $\square_{κ, 2}$ may not be strengthened to $\square_κ$. Using methods of Golshani and Hayut we obtain our consistency result simultaneously for all regular $κ$.
Almost Kurepa Suslin trees and destructibility of the Guessing Model Property
本文证明了在ω₂处猜测模型原理与几乎Kurepa Suslin树存在的一致性，并展示了该原理可能被cccforcing破坏，同时探讨了弱Kurepa树与树性质的关系。
In this note, we investigate a new model theoretical tree property, called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and $k$-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure group, pure fields, and valued fields. More precisely, we prove Mekler's construc…
For an inaccessible cardinal $κ$, the super tree property (ITP) at $κ$ holds if and only if $κ$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit of $ω$ many supercompact cardinals. Then we show that it can consistently hold at $\aleph_{ω+1}$. We also consider a stronger principle, ISP, and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining…
A preservation theorem for theories without the tree property of the first kind
本文证明了几何理论的NTP$_1$性质在可爱对和$H$-结构中保持，并给出非简单几何NTP$_1$理论的例子。
We prove that successors of singular limits of strongly compact cardinals have the strong tree property. We also prove that aleph_{omega+1} can consistently satisfy the strong tree property.
We consider global analogues of model-theoretic tree properties. The main objects of study are the invariants related to Shelah's tree property $κ_{\text{cdt}}(T)$, $κ_{\text{sct}}(T)$, and $κ_{\text{inp}}(T)$ and the relations that obtain between them. From strong colorings, we construct theories $T$ with $κ_{\text{cdt}}(T) > κ_{\text{sct}}(T) + κ_{\text{inp}}(T)$. We show that these invariants have distinct structural consequences, by investigating the decay of saturation in ultrapowers of models of $T$, where $T$ is some theory with $κ_{\text{cdt}}(T)$, $κ_{\text{sct}}(T)$, or $κ_{\text{inp…
The main results of this note are: It is consistent that every subparacompact space $X$ of size $ω_1$ is a $D$-space; If there exists a Michael space, then all productively Lindelöf spaces have the Menger property, and, therefore, are $D$-spaces; and Every locally $D$-space which admits a $σ$-locally finite cover by Lindelöf spaces is a $D$-space.
Building on previous work of [BPS] we investigate $σ$-closed partial orders of size continuum. We provide both an internal and external characterization of such partial orders by showing that (1) every $σ$-closed partial order of size continuum has a base tree and that (2) $σ$-closed forcing notions of density $\mathfrak c$ correspond exactly to regular suborders of the collapsing algebra $Coll(ω_1, 2^ω)$. We further study some naturally ocurring examples of such partial orders.
In this note we present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hernández-Gutiérrez and Hrušák. The method of the proof also allows us to obtain a metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.
No convenient internal characterization of spaces that are productively Lindelof is known. Perhaps the best general result known is Alster's internal characterization, under the Continuum Hypothesis, of productively Lindelof spaces which have a basis of cardinality at most $\aleph_1$. It turns out that topological spaces having Alster's property are also productively weakly Lindelof. The weakly Lindelof spaces form a much larger class of spaces than the Lindelof spaces. In many instances spaces having Alster's property satisfy a seemingly stronger version of Alster's property and consequently …
Strong Tree Properties Along Many Segments of Successors of Singulars
该研究从大基数出发，通过力迫构造证明了ITP可在奇异基数的可数初始段后继处同时成立，并推广得到奇异基数多共尾度长段后继处的强树性质的一致性。
The special Aronszajn tree property at $\aleph_2$ and $GCH$
本文从弱紧基数出发构建了一个满足 $GCH$ 的通用扩张，证明了所有 $\aleph_2$-Aronszajn 树均是特殊的且不存在 $\aleph_2$-Souslin 树，从而回答了1970年代的著名开放问题。
Strong tree properties, Kurepa trees, and guessing models
本文研究广义树性质与猜测模型性质，提出几乎猜测性质以刻画纤细树性质，并证明其推论、在 Mitchell 模型中的成立性及相关一致性结果。
Let n be an integer greater than 1. A tree T is an n-ary tree provided that every node in T has at most n immediate successors. A forcing notion P has the n-localization property if every function from omega to omega in an extension via P is an omega-branch in an n-ary tree from the ground model. In the present paper we are interested in getting the n-localization property for countable support iterations.
In the first part of the paper, we show that if $ω\le κ< λ$ are cardinals, $κ^{<κ} = κ$, and $λ$ is weakly compact, then in $V[\M(κ,λ)]$ the tree property at $λ= κ^{++V[\M(κ,λ)]}$ is indestructible under all $κ^+$-cc forcing notions which live in $V[\Add(κ,λ)]$, where $\Add(κ,λ)$ is the Cohen forcing for adding $λ$-many subsets of $κ$ and $\M(κ,λ)$ is the standard Mitchell forcing for obtaining the tree property at $λ= (κ^{++})^{V[\M(κ,λ)]}$. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that $λ$ is superco…
Iwasa investigated the preservation of various covering properties of opological spaces under Cohen forcing. By improving the argument in Iwasa's paper, we prove that the Rothberger property, the Menger property and selective screenability are also preserved under Cohen forcing and forcing with the measure algebra.
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 a…
We prove that a strong version of Chang's Conjecture, equivalent to the Weak Reflection Principle at $ω_2$, together with $2^ω=ω_2$, imply there are no $ω_2$-Aronszajn trees.
Tree Properties at Successors of Singulars of Many Cofinalities
本研究利用多个超紧基数，证明树性质可在多个不同共尾性的奇异基数小后继上相容成立，构造了特定基数上树性质成立的模型，并将结果推广到强树性质及不可数长共尾序列情形。
According to a result of Kocinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals $X$ satisfies the Hurewicz property if, and only if, each large open cover of $X$ contains a groupable subcover. This solves in the affirmative a problem of Scheepers. The proof uses a rigorously justified a…
The_additivity_number_ of a topological property (relative to a given space) is the minimal number of subspaces with this property whose union does not have the property. The most well-known case is where this number is greater than Aleph_0, i.e. the property is sigma-additive. We give a rather complete survey of the known results about the additivity numbers of a variety of topological covering properties, including those appearing in the Scheepers diagram (which contains, among others, the classical properties of Menger, Hurewicz, Rothberger, and Gerlits-Nagy). Some of the results proved her…
In this paper we prove the equiconsistency of ``Every omega_1 tree which is first order definable over H_{omega_1} has a cofinal branch'' with the existence of a Pi^1_1 reflecting cardinal. The proof uses a definable version of Ramsey theorem on aleph_1 which is again equiconsistent with a Pi^1_1 reflecting cardinal. We also prove that the addition of $MA$ to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.
We prove that the Hurewicz property is not preserved by finite products in the Miller model. This is a consequence of the fact that Miller forcing preserves ground model $γ$-spaces.
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