A partition relation using strongly compact cardinals

具有可数划分的Dedekind有限基数

Dedekind-finite cardinals having countable partitions

本文研究没有可数子集但在两种满射意义上非“surjectively Dedekind finite”的集合可能具有的结构。

On $ω$-Strongly Measurable Cardinals

We prove several consistency results concerning the notion of $ω$-strongly measurable cardinal in HOD. In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(κ) = κ$, that every successor of a regular cardinal is $ω$-strongly measurable in HOD.

Suslin trees, the bounding number, and partition relations

We investigate the unbalanced ordinary partition relations of the form $λ\rightarrow {(λ, α)}^{2}$ for various values of the cardinal $λ$ and the ordinal $α$. For example, we show that for every infinite cardinal $κ,$ the existence of a $κ^{+}-$Suslin tree implies $κ^{+} \not\rightarrow {\left( κ^{+}, {\log}_κ(κ^{+}) + 2 \right)}^{2}$. The consistency of the positive partition relation $\mathfrak{b} \rightarrow {(\mathfrak{b}, α)}^{2}$ for all $α< ω_{1}$ for the bounding number $\mathfrak{b}$ is also established from large cardinals.

On the cofinality of the least $λ$-strongly compact cardinal

In this paper, we characterize the possible cofinalities of the least $λ$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $δ$, that carries a $λ$-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above $δ$, that the least $λ$-strongly compact cardinal has cofinality $δ$. On the other hand, provably the cofinality of the least $λ$-strongly compact cardinal always carries a $λ$-complete uniform ultrafilter.

Ramsey theory over partitions I: Positive Ramsey relations from forcing axioms

In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over them is uncovered. At the level of the first uncountable cardinal this gives rise to a duality theorem under Martin's Axiom: a function $p:[ω_1]^2\rightarrowω$ witnesses a weak negative Ramsey relation when $p$ plays the role of a coloring if and only if a positive Ramsey relation holds over $p$ when $p$ plays the role of a partition. The consistency of posi…

Positive partition relations for P_{kappa,lambda}

Let kappa a regular uncountable cardinal and lambda a cardinal >kappa, and suppose lambda^{<kappa} is less than the covering number for category cov(M_{kappa,kappa}). Then (a) I_{kappa,lambda}^+ -->^kappa (I_{kappa, lambda}^+,omega +1)^2, (b) I_{kappa, lambda}^+ -->^kappa [I_{kappa, lambda}^+]_{kappa^+}^2 if kappa is a limit cardinal, and (c) I_{kappa, lambda}^+ -->^kappa (I_{kappa, lambda}^+)^2 if kappa is weakly compact.

Exactly controlling the non-supercompact strongly compact cardinals

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify previous results of the first author.

Diamond (on the regulars) can fail at any strongly unfoldable cardinal

If kappa is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which Diamond_kappa(REG) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

Strongly compact cardinals and the continuum function

We study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals.

大基数上的Galvin性质及其在划分演算中的应用

Galvin's property at large cardinals and an application to partition calculus

本文研究大基数上不具有Galvin性质的κ-完全超滤子的一致性，证明基模型超滤子既可扩张为非Galvin超滤子，也可扩张为满足Galvin性质的P-点超滤子，并由此在划分演算问题λ→(λ,ω+1)^2中获得一致的新实例。

On special partitions of metric spaces

The main result of this paper is to show that, if $κ$ is the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$, then there exists a complete metric space of cardinality not greater than $ 2^κ$ admitting a Kuratowski partition.

Strongly compact cardinals and ordinal definability

This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the fi…

Structure theorems above a strongly compact cardinal

While many inner model theoretic combinatorial principles are incompatible with large cardinal axioms, on some rare occasions, large cardinals actually imply that the structure of the universe of sets is analogous to the canonical inner models. This note provides two new instances of this phenomenon.

Easton's Theorem for Ramsey and Strongly Ramsey cardinals

We show that, assuming GCH, if $κ$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $κ$ and obeying the constraints of Easton's theorem, namely, $F(α)\leq F(β)$ for $α\leqβ$ and $α<\cf(F(α))$, then there is a cofinality preserving forcing extension in which $κ$ remains Ramsey or strongly Ramsey respectively and $2^δ=F(δ)$ for every regular cardinal $δ$.

强紧基数之上的广义连续统假设

The generalized continuum hypothesis above a strongly compact cardinal

本文证明若κ为强紧基数且GCH在κ之下成立，则GCH整体成立，解答了Woodin的长期开放问题。

Strongly uplifting cardinals and the boldface resurrection axioms

We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.

On shattering, splitting and reaping partitions

In this article we investigate the dual-shattering cardinal H, the dual-splitting cardinal S and the dual-reaping cardinal R, which are dualizations of the well-known cardinals h (the shattering cardinal, also known as the distributivity number of P(omega) modulo finite, s (the splitting number) and r (the reaping number). Using some properties of the ideal J of nowhere dual-Ramsey sets, which is an ideal over the set of partitions of omega, we show that add(J)=cov(J)=H. With this result we can show that H > omega_1 is consistent with ZFC and as a corollary we get the relative consistency of H…

Inner models with large cardinal features usually obtained by forcing

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κfor which 2^κ=κ^+, another for which 2^κ=κ^++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable c…

A family of covering properties for forcing axioms and strongly compact cardinals

This paper presents the main results in my Ph.D. thesis. In what follows several proofs of SCH are presented introducing a family of covering properties which implies both SCH and the failure of various forms of square. These covering properties are also applied to investigate models of strongly compact cardinals or of strong forcing axioms like MM or PFA.

方括号划分关系的一致性

Consistency of square bracket partition relation

本文改进了方括号划分关系的一致性结果，展示了连续统可以较小且不依赖大基数，同时涉及的基数在forcing后仍小于连续统。

Successors of Singular Cardinals and Coloring Theorems II

We investigate negative square-brackets partition relations at successors of singular cardinals of countable cofinality. Along the way we prove some club-guessing results.

Combinatorial dichotomies and cardinal invariants

Assuming the P-ideal dichotomy, we attempt to isolate those cardinal characteristics of the continuum that are correlated with two well-known consequences of the proper forcing axiom. We find a cardinal invariant $\mathfrak{x}$ such that the statement that $\mathfrak{x} > ω_{1}$ is equivalent to the statement that 1, $ω$, $ω_{1}$, $ω\times ω_{1}$, and ${\left[ω_{1}\right]}^{< ω}$ are the only cofinal types of directed sets of size at most ${\aleph}_{1}$. We investigate the corresponding problem for the partition relation $ω_{1} \rightarrow (ω_{1}, α)^2$ for all $α< ω_{1}$. To this effect, we i…

Superatomic Boolean algebras constructed from strongly unbounded functions

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $κ,λ$ are infinite cardinals such that $κ^{+++} \leq λ$, $κ^{<κ}=κ$ and $2^κ= κ^+$, and $η$ is an ordinal with $κ^+\leq η<κ^{++}$ and $cf(η) = κ^+$. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra $B$ such that - $ht(B) = η+ 1$, - the cardinality of the $α$th level of $B$ is $κ$ for every $α<η$, - and the cardinality of the $η$th level of $B$ is $λ$ Especially, $\<ω\>_{ω_1}\concatenation \<ω_3\>$ and $\<ω_1\>_{ω_2}\concatenation \<ω_4\>$ can be cardi…

一个集合的具有 $n$ 个非单例块的有限划分

The finitary partitions with $n$ non-singleton blocks of a set

本文在ZF公理系统下研究无限集合的有限划分，证明了关于非单例块划分的基数幂等式的两个定理，并构造了一个无限基数使得相关幂基数严格递增。

Patterns of Compact Cardinals

We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V is a model of ``ZFC + Omega is the least inaccessible limit of measurable limits of supercompact cardinals + f :Omega-->2 is a function'', then there is a partial ordering P in V so that for Vbar = V^P, Vbar_Omega models ``ZFC + There is a proper class of compact cardinals + If f(alpha) = 0, then the alpha-th compact cardinal isn't supercompact + If f(alpha) = 1, then t…

大基数层级中的结构反射模式

Patterns of Structural Reflection in the large-cardinal hierarchy

该研究揭示首个可测基数以下大基数层级的结构反射新模式，刻画强不可展开与微妙基数，引入$C^{(n)}$-强不可展开基数并证明其形成自然层级，证实大基数低区域与更高区域结构完全类似。

通过紧性对列表着色的大基数刻画

Large cardinal characterizations via compactness for list colourings

该研究引入含知名大基数概念的紧性基数分层，通过其刻画弱紧等大基数，并得到新紧性基数大小的下界。

Tameness from Large Cardinal Axioms

We show that Shelah's Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with $LS(K)$ below a strongly compact cardinal $κ$ is $< κ$ tame and applying the categoricity transfer of Grossberg and VanDieren. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called \emph{type shortness}, and show that it follows simila…

Jonsson 滤子与 Magidor 滤子

Jonsson and Magidor filters

本文在AD框架下研究方括号划分关系的滤子版本，证明强划分基数之上Kleinberg序列的奇异基数非Magidor但序列极限是，并通过力迫得到带Magidor滤子的奇异基数。

A note on infinite partitions of free products of Boolean algebras

If $A$ is an infinite Boolean algebra the cardinal invariant $\mathfrak{a}(A)$ is defined as the smallest size of an infinite partition of $A$. The cardinal $\mathfrak{a}(A\oplus B)$, where $A\oplus B$ is the free product of the Boolean algebras $A$ and $B$ (whose dual topological space is the product of the dual topological spaces of $A$ and $B$), is below both $\mathfrak{a}(A)$ and $\mathfrak{a}(B)$. The equality $\mathfrak{a}(A\oplus B)=\min\lbrace\mathfrak{a}(A),\mathfrak{a}(B)\rbrace$ is not known to hold for all infinite Boolean algebras $A$ and $B$. Here some lower bounds of $\mathfrak{…

适用于强逻辑的弱紧基数与序数类的微妙性质

Weak compactness cardinals for strong logics and subtlety properties of the class of ordinals

该研究通过序数类组合性质刻画抽象逻辑弱紧基数存在性，证明其不蕴含强不可达基数存在，并给出其与真类微妙基数的一致性等价条件及HOD内模型中的相关推论。

A note on strong negative partition relations

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.

Some combinatorial properties of Ultimate L and V

This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate \(L\), providing some evidence for the Ultimate \(L\) Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality…

A partition theorem for scattered order types

If phi is a scattered order type, mu a cardinal, then there exists a scattered order type psi such that psi->[phi]^1_{mu,aleph_0} holds.

Weakly extendible cardinals and compactness of extended logics

We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located strictly between that of strongly unfoldable (i.e. shrewd) cardinals, and strongly uplifting cardinals. Weak compactness of many other logics can be connected to certain variants of the notion of weakly extendible cardinals. We also show that, under V=L, a cardinal $κ$ is the weak compactness number of ${\cal L}^{\aleph_0,II}_{stat,κ,ω}$ if and only if it is th…

A note on $δ$-strongly compact cardinals

In this paper we investigate more characterizations and applications of $δ$-strongly compact cardinals. We show that, for a cardinal $κ$ the following are equivalent: (1) $κ$ is $δ$-strongly compact, (2) For every regular $λ\ge κ$ there is a $δ$-complete uniform ultrafilter over $λ$, and (3) Every product space of $δ$-Lindelöf spaces is $κ$-Lindelöf. We also prove that in the Cohen forcing extension, the least $ω_1$-strongly compact cardinal is a precise upper bound on the tightness of the products of two countably tight spaces.

Strong Tree Properties at Successors of Singular Cardinals

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.

A note on cardinal preserving embeddings

We prove the consistency of the theory ZFC + there is a strongly compact cardinal from the existence of a cardinal preserving embedding from the universe into an inner model. The proof almost shows that under SCH, every cardinal preserving embedding preserves the cofinality and continuum functions.

关于$\mathbb{P}_{max}$扩张中的$\omega$-强可测基数

On $ω$-strongly measurable cardinals in $\mathbb{P}_{\max}$ extensions

本文证明了在特定Chang型决定性模型的$\mathbb{P}_{max}$扩张中，特定基数上的club filter限制到HOD上构成超滤，从而回答了Ben-Neria和Hayut提出的问题4.11。

A partition theorem for a large dense linear order

Let Q_K=(Q,<_Q)$ be a strongly K-dense linear order of size K for a suitable cardinal K. We prove, for all integers m > 1 that there is a finite value t_m^+ such that the set of all m-tuples from Q can be divided into t_m^+ many classes, such that whenever any of these classes C is colored with fewer than K many colors, there is a copy Q* of Q_K such that all m-tuples from Q* in C receive the same color. As a consequence we obtain that whenever we color the m-tuples of Q with fewer than K many colors, there is a copy of Q_K all m-tuples from which are colored in at most t_m^+ colors. In other …

The tree property at successors of singular cardinals

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.

The Lottery Preparation

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal kappa, for example, becomes fully indestructible by kappa-directed closed forcing; a strong cardinal kappa becomes indestructible by less-than-or-equal-kappa-strategically closed forcing; and a strongly compact cardinal kappa becomes indestructible …

Collapsing the cardinals of $HOD$

Assuming that $GCH$ holds and $κ$ is $κ^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $κ$ remains strongly inaccessible and $(α^+)^{HOD} < α^+$ for every infinite cardinal $α< κ$. In particular the rank-initial segment $W_κ$ is a model of ZFC in which $(α^+)^{HOD} < α^+$ for every infinite cardinal $α$.

没有弱 Kurepa 树的强非饱和 Aronszajn 树

A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees

本文基于 Chang 猜想的否定与不可达基数等大基数假设，通过构造 c.c.c. 和 Y-proper 等力迫偏序集，证明了强非饱和 Aronszajn 树的存在性与弱 Kurepa 树的不存在性及相关双基数树性质保持一致。

Strong unfoldability, shrewdness and combinatorial consequences

We show that the notions of "strongly unfoldable cardinals", introduced by Villaveces in his model-theoretic studies of models of set theory, and "shrewd cardinals", introduced by Rathjen in a proof-theoretic context, coincide. We then proceed by using ideas from the proof of this equivalence to establish the existence of "ordinal anticipating Laver functions" for strong unfoldability. With the help of these functions, we show that the principle $\Diamond_κ(\mathrm{Reg})$ holds at every strongly unfoldable cardinal $κ$ with the property that there exists a subset $z$ of $κ$ such that every sub…

在所有不可数基数上平方失败比Woodin极限的Woodin基数更弱

The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals

该研究通过强制选择公理，在满足ZF的最小初始段的Nairian模型中展示所有不可数基数上平方失败，并表明正则基数在HOD中为omega-强可测，从而揭示了HOD假设在ZFC中的不可证性。

Compactness and Comparison

Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we characterize strongly compact cardinals in terms of supercompactness: a strongly compact cardinal is either supercompact or a limit of supercompact cardinals. Assuming the Ultrapower Axiom and the GCH, we also prove a local result that roughly states that every countably complete ultrafilter factors as a finite iteration of supercompact ultrafilters.

A class of strong diamond principles

In the context of large cardinals, the classical diamond principle Diamond_kappa is easily strengthened in natural ways. When kappa is a measurable cardinal, for example, one might ask that a Diamond_kappa sequence anticipate every subset of kappa not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function l:kappa-->V_kappa such that for any A in H(kappa+) there is an embedding j:V-->M having critical point kappa with j(l)(kappa)=A. This and similar principles formulated for many other large cardinal notions, including weakly compact, i…

A note on highly connected and well-connected Ramsey theory

We study a pair of weakenings of the classical partition relation $ν\rightarrow (μ)^2_λ$ recently introduced by Bergfalk-Hrušák-Shelah and Bergfalk, respectively. Given an edge-coloring of the complete graph on $ν$-many vertices, these weakenings assert the existence of monochromatic subgraphs exhibiting high degrees of connectedness rather than the existence of complete monochromatic subgraphs asserted by the classical relations. As a result, versions of these weakenings can consistently hold at accessible cardinals where their classical analogues would necessarily fail. We prove some complem…

Gap forcing: generalizing the Levy-Solovay theorem

The landmark Levy-Solovay Theorem limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.

The Halpern--Läuchli Theorem at singular cardinals and failures of weak versions

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…

Cichon's Diagram for uncountable cardinals

We develop a version of Cichon's diagram for cardinal invariants on the generalized Cantor space 2^kappa or the generalized Baire space kappa^kappa where kappa is an uncountable regular cardinal. For strongly inaccessible kappa, many of the ZFC-results about the order relationship of the cardinal invariants which hold for omega generalize; for example we obtain a natural generalization of the Bartoszynski-Raisonnier-Stern Theorem. We also prove a number of independence results, both with <kappa-support iterations and kappa-support iterations and products, showing that we consistently have stri…

Weak prediction principles

We prove the consistency of the failure of the weak diamond $Φ_λ$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $Ψ_λ$ is equivalent to the statement $2^{<λ}<2^λ$ and hence holds at every strongly inaccessible cardinal.

Transformations of the transfinite plane

We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every inaccessible cardinal $κ$, if $κ$ admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation $κ\nrightarrow[κ]^2_κ$ implies that for every Abelian group $(G,+)$ of size $κ$, there exists a map $f:G\rightarrow G$ such that, for every $X\subseteq G$ of size $κ$ and every $g\in G$, there exis…