Shimon Garti
Erdos, Hajnal and Rado asked whether $\binom{\aleph_{ω_1}}{\aleph_2}\rightarrow\binom{\aleph_{ω_1}}{\aleph_0}_2$ and whether $\binom{\aleph_{ω_1}}{\aleph_2}\rightarrow\binom{\aleph_{ω_1}}{\aleph_1}_2$. We prove that both relations are independent over ZFC. We shall also prove that $\binomμ{\aleph_2}\rightarrow\binomμ{\aleph_2}_2$ is independent over ZF for some $μ$ of cofinality $ω_1$.
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May the successor of a singular cardinal be Jonsson?
该研究为数学逻辑领域的综述工作，汇总了奇异基数的后继成为Jonsson基数的必要条件，为相关大基数问题研究提供参考。
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.
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 investigate negative square-brackets partition relations at successors of singular cardinals of countable cofinality. Along the way we prove some club-guessing results.
We introduce a combinatorial notion of measures called Rudin-Keisler capturing and use it to give a new construction of elementary substructures around singular cardinals. The new construction is used to establish mutual stationary results at the first successor of singular cardinals $\langle \aleph_{ωn + 1}\rangle_{n< ω}$.
We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413] (math.LO/9809199), but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.
Assuming that $0^\dagger$ does not exist, we prove that if there is a partition of $\mathbb R$ into $\aleph_ω$ Borel sets, then there is also a partition of $\mathbb R$ into $\aleph_{ω+1}$ Borel sets.
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.
Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection
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…
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…
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 $κ^{++}$.
We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size…
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 paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of cofinality aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such mu there are mu^{++} graphs on mu^+ that taken jointly are universal for the graphs on mu^+, while 2^{mu^+}>> mu^{++} . The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal kappa exists. The result on the exist…
Total Failure of Approachability at Successors of Singulars of Countable Cofinality
在可数余数的后继点中，研究了非可接近点的存在性，解决了Mitchell和Foreman、Shelah的问题。
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。
We prove the consistency of a strong polarized relation for a cardinal and its successor, using pcf and forcing
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。
Failure of Approachability at the Successor of the first Singular for any Cofinality
该研究解决了两个关于ℵ_{ω+1}组合学的长期公开问题，证明对任意n≥1，GCH成立且存在共尾性ℵ_n的驻集不在可逼近性理想中，并得到良态性与可逼近性在共尾性ℵ_1的驻点上不同的模型。
We strengthen non-structure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fraisse games between a fixed group of cardinality lambda and a free Abelian group. A group is called epsilon-game-free if the isomorphism player has a winning strategy in the game (of the described form) of length epsilon in lambda. We prove for a large set of successor cardinals lambda=mu^+ existence of nonfree (mu*omega_1)-game-free groups of cardinality lambda. We concentrate on successors of singular cardinals.
The notion of stationary reflection is one of the most important notions of combinatorial set theory. We investigate weak reflection, which is, as the name suggests, a weak version of stationary reflection. This sort of reflection was introduced in [DjSh:545] (math.LO/9601219), where it was shown that weak reflection has applications to various guessing principles, in the sense that if there is no weak reflection, than a guessing principle holds, and an application dealing with the saturation of normal filters. Further investigations of weak reflection were carried in [CDSh:571] (math.LO/95042…
Let $\mathrm{cof}(μ)=μ$ and $κ$ be a supercompact cardinal with $μ<κ$. Assume that there is an increasing and continuous sequence of cardinals $\langleκ_ξ\mid ξ<μ\rangle$ with $κ_0:=κ$ and such that, for each $ξ<μ$, $κ_{ξ+1}$ is supercompact. Besides, assume that $λ$ is a weakly compact cardinal with $\sup_{ξ<μ}κ_ξ<λ$. Let $Θ\geqλ$ be a cardinal with $\mathrm{cof}(Θ)>κ$. Assuming the $\mathrm{GCH}_{\geqκ}$, we construct a generic extension where $κ$ is strong limit, $\mathrm{cof}(κ)=μ$, $2^κ= Θ$ and both $\mathrm{TP}(κ^+)$ and $\mathrm{TP}(κ^{++})$ hold. Further, in this model there is a very …
We formulate and prove (in {\sf ZFC}) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle $Pr_1(μ^+,μ^+,μ^+,\cf(μ))$ for singular $μ$.
We prove that $α_M(λ)$ can be successor of a supercompact cardinal, when $λ$ is a Magidor cardinal. From this result we obtain the consistency of $α_M(λ)$ being a successor of a singular cardinal with uncountable cofinality.
Negating the Galvin Property
本文证明了 Galvin 性质在强极限奇异基数及一般奇异基数的后继处可一致地不成立，并分析了 Prikry 型力迫对其强失败的影响及大基数背景下的更强形式。
Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{μ^{++}}{μ^{+}}\to\binomκ{μ^{+}}_μ$ and $\mathrm{Tr}_{\mathrm{Chr}}(μ^{+++},μ^{+})$ in the extension by Prikr…
Superclub, splitting, separating statements
本文证明 superclub 蕴含分裂数等于最小不可数基数，并在弱紧、超紧及正则、奇异基数后继处利用 Galvin 性质分离 tiltan 与 superclub。
Club Stationary Reflection and other Combinatorial Principles at $\aleph_{ω+2}$
本文研究可数共尾性奇异基数双后继上的树性质与俱乐部驻留反射，构造了满足这些性质的模型，并证明一类满足关联假设的偏序集保持俱乐部驻留反射。
We establish a coloring theorem for successors of a singular cardinals, and use it prove that for any such cardinal $μ$, we have $μ^+\nrightarrow[μ^+]^2_{μ^+}$ if and only if $μ^+\nrightarrow[μ^+]^2_θ$ for arbitrarily large $θ<μ$.
We obtain strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. Along the way, we establish new results in club-guessing and in the general theory of ideals.
1. For many regular cardinals lambda (in particular, for all successors of singular strong limit cardinals, and for all successors of singular omega-limits), for all n in {2,3,4, ...} : There is a linear order L such that L^n has no (incomparability-)antichain of cardinality lambda, while L^{n+1} has an antichain of cardinality lambda . 2. For any nondecreasing sequence (lambda2,lambda3, ...) of infinite cardinals it is consistent that there is a linear order L such that L^n has an antichain of cardinality lambda_n, but not one of cardinality lambda_n^+ .
We give two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular and at least equal to aleph_3, while the second is a negative consistency result which applies to the situation of lambda a singular strong limit with 2^lambda>lambda^+. The first result shows that in ZFC there is a guessing of unbounded subsets of S^{lambda^+}_lambda. The second result is a consistency result (assuming a supercompact cardinal exists) showing that a natural guessing fails. A result of Shelah in math.LO/9808140 shows that if 2^lambda=lambd…
We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem: Suppose $κ$ is a singular strong limit cardinal and $2^κ>= λ$ where $λ$ is not the successor of a cardinal of cofinality at most $κ$. (i) If $\cofinality(κ)>\gw$ then $o(κ)\geλ$. (ii) If $\cofinality(κ)=\gw$ then either $o(κ)\geλ$ or $\set{\ga:K\sat o(\ga)\ge\ga^{+n}}$ is cofinal in $κ$ for each $n\in\gw$. In order to prove this theorem we give a detailed analysis of the sequences of indiscernibles which come from appl…
Quadruples and cubes
本文在ZFC中证明阶梯立方体与四元组关系的失败条件，并利用奇妙理想建立后继基数上的正极化关系，同时否定连续基数上的此类理想存在。
Let $κ$,$λ$ be regular uncountable cardinals such that $λ> κ^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(κ) = λ$ starting from a ground model in which $o(κ) = λ$ and prove that assuming $\neg 0^¶$, $s(κ) = λ$ implies that $o(κ) \geq λ$ in the core model.
Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines
结合序数上的行走、club guessing和强着色等组合方法，证明了若存在$\aleph_2$-Aronszajn线则存在不含$\aleph_2$-Countryman线的特例，解决了Moore的问题，并将相关非结构定理推广至多种基数情形。
Stationary reflection using ancestrally forced conditions
该研究提出新方法，从κ⁺-超紧基数出发，构建奇异基数后继处平稳反射模型，得到ℵ_{ω₁+1}处可数同时反射的模型。
The main result is that for lambda strong limit singular failing the continuum hypothesis (i.e. 2^lambda > lambda^+), a polarized partition theorem holds.
We analyze a natural function definable from a scale at a singular cardinal, and using this function we are able to obtain quite strong negative square-brackets partition relations at successors of singular cardinals. The proof of our main result makes use of club-guessing, and as a corollary we obtain a fairly easy proof of a difficult result of Shelah connecting weak saturation of a certain club-guessing ideal with strong failures of square-brackets partition relations. We then investigate the strength of weak saturation of such ideals and obtain some results on stationary reflection.
We prove the consistency of $\binom{μ^+}μ\nrightarrow\binom{μ^+ ω_1}{μ μ}$ where $μ$ is a strong limit singular cardinal of countable cofinality. This result can be forced at limit of measurable cardinals and at small cardinals like $\aleph_ω$.
This paper continues a line of investigation of the Halpern--Läuchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--Läuchli Theorem for one tree of height $κ$ holds whenever $κ$ is strongly inaccessible and the coloring takes less than $κ$ colors. We prove consistency of the Halpern--Läuchli Theorem for finitely many trees of height $κ$, where $κ$ is a strong limit cardinal of countable cofinality. On the other hand, we prove failure of weak forms of Halpern--\Lauchli\ for trees of height $κ$, whenever $κ$ is a strongly inaccessible, non-Mahlo cardinal or a singular strong…
Remarks on Halin's end-degree Conjecture
本文在ZFC框架下证明了Halin端度猜想的新实例，并探讨了其与奇异基数假设的关系，揭示了该猜想在特定模型中的失败情况，提供了新的独立性结果。
We list some open problems, concerning the polarized partition relation. We solve a couple of them by showing that for every singular cardinal $μ$ one can force the strong polarized relation with respect to the pair $μ^+,μ$.
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