Haim Horowitz, Saharon Shelah
We construct a Borel graph G such that ZF+DC+"There are no maximal independent sets in G" is equiconsistent with ZFC+"There exists an inaccessible cardinal".
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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.
We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality $\mathfrak p\leq\mathfrak b$ does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting numbers. Lastly, we show that the inclusion ordering on $\mathcal P(ω)…
Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number $\mathfrak b$. We analyze some of the basic behavior of these properties, showing for instance that the property corresponding to the splitting …
Borel complexity of families of finite equivalence relations via large cardinals
该研究在假设ω-Erdos基数存在的前提下，判定有限等价类的等价关系理论的Borel完全性，构建了用于界定复杂度和判定完全性的工具，并分类了细化等价关系理论的归约。
We prove that consistently, cov($\mathcal{M})< λ_\mathbf{0} < λ_\mathbf{1} < λ_\mathbf{\infty} < 2^{\aleph_0}$, where $λ_\mathbf{0}$ denotes the weak Borel chromatic number of the Kechris-Solecki-Todorčević graph $\mathbb{G}_0$, that is, the minimal cardinality of a $\mathbb{G}_0$-independent Borel covering of $2^ω$, while $λ_\mathbf{1}$ and $λ_\infty$ are the corresponding invariants of the graph $\mathbb{G}_1$ and the simple graph associated with the equivalence relation $E_0$.
We construct a Borel maximal eventually different family.
Generalized Borel Sets
本文在类波兰空间框架下研究广义波莱尔分层，给出κ⁺-波莱尔分层性质与非坍塌条件，揭示奇异基数下κ-波莱尔分层的独特性，并解决正则基数下相关问题。
On relatively analytic and Borel subsets
定义z为函数f：X-> Y的最小基数，其中X和Y是实数集，使得没有Borel函数g扩展f。在本文中，我们证明了它是相对一致的ZFC有b <z，其中b是，通常情况下，最小的基数的一个无界的家庭在w ^w。这回答了Zapletal提出的一个问题。我们还证明了存在一个实数集X使得X的Borel序是有界的，但存在X的一个相对解析子集，它不是相对余解析的，这与ZFC是相对一致的.这回答了莫尔丁的一个问题。
Let $Γ$ be a countable group. A classical theorem of Thorisson states that if $X$ is a standard Borel $Γ$-space and $μ$ and $ν$ are Borel probability measures on $X$ which agree on every $Γ$-invariant subset, then $μ$ and $ν$ are equidecomposable, i.e. there are Borel measures $(μ_γ)_{γ\inΓ}$ on $X$ such that $μ= \sum_γμ_γ$ and $ν= \sum_γγμ_γ$. We establish a generalization of this result to cardinal algebras.
In $\mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E \in L(\mathbb{R})$ on $\mathbb{R}$ with all $\mathbfΔ_1^1$ classes and every $σ$-ideal $I$ on $\mathbb{R}$ so that the associated forcing $\mathbb{P}_I$ of $I^+$ $\mathbfΔ_1^1$ subsets is proper, there exists some $I^+$ $\mathbfΔ_1^1$ set $C$ so that $E \upharpoonright C$ is a $\mathbfΔ_1^1$ equivalence relation. In $\mathsf{ZF} + \mathsf{DC} + \mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R}))$, for every equivalence relation $E$ on $\mathbb{R}$ with…
We show there is a closed (in fact effectively closed, i.e., $Π^0_1$) eventually different family (working in ZF or less).
We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $κ$, let {\sf BC}$_κ$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_κ$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. (…
Borel Complexity and the Schröder-Bernstein Property
本研究提出Borel可归约性的新不变量“厚度”，证明挠阿贝尔群的所有Friedman-Stanley跳跃非Borel完全，并在大基数假设下证明具有Schröder-Bernstein性质的$\u27e8\ucf79_{\u03c9_1 \u03c9}$语句非Borel完全。
Borel images of sets of reals
本文的主要目的是把有关基数不变量的几个结果推广到有关相关集合族的陈述中。我们还讨论了集合的可加性与其Borel象之间的关系。最后，我们提出的最小集，这是不是强烈微薄的大小估计。
Proper classes of maximal $θ$-independent families from large cardinals
该研究拓展Kunen的力迫技术，分别利用单个θ^+-强紧基数的集合泛型扩张、带真类可测基数模型的类泛型扩张，构造不同不可数θ下的极大θ-无关族真类，推进大基数与组合集合论交叉研究。
Potential Wadge classes
设$\bfΓ$是Borel类或Borel集的Wadge类，而$2\leq d\leqω$是基数。我们研究了${\mathbb R}^d$的Borel子集，通过在实直线上改进波兰拓扑，可以使之成为$\bfΓ$。这些集合称为潜在$\bfΓ$。我们给出了一个测试来识别潜在的$\bfΓ$集。
A Borel maximal cofinitary group
本文构造了一个 Borel 极大余有限群，解决了描述集合论与群论交叉领域中的构造性问题。
A nonhereditary Borel-cover gamma-set
本文证明了：如果存在连续统上的Borel-覆盖Gamma-集的基数，则存在一个不遗传的Borel-覆盖Gamma-集。实数集X是Borel-覆盖Gamma-集当且仅当对于每一个可数的Borel集族，若它是一个omega-覆盖，则包含一个Gamma-覆盖。这也表示为S_1（Borel_omega，Borel_gamma）。这个结果部分回答了Bartoszynski和Tsaban的一个问题。Tsaban指出，它还给出了一个集的例子，这是一个伽玛集和西格玛集，但不是遗传伽玛，这回答了一个问题的Bukovsky，Reclaw和Repicky。
For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that |(eta_alpha+B) cap (eta_beta+B)|>5 for all alpha,beta<lambda, but (2) there is no perfect set of such eta's. The construction closely follows the one from Shelah math.LO/9802134
The Necessary Maximality Principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal
c.c.c.c.的必要极大性原理强迫断言任何关于c.c.c.在进一步的c.c.c. c中可能成为真的扩展。并在所有后续c.c.c.扩展，在包含实数的最小扩展中已经为真。我们证明了这个原理与弱紧基数的存在性是等相容的。
Borel sets with large squares
本文修正了早期研究，给出基数相关的充分条件$\oplus_μ$，证明其与Borel集大正方形存在性、无穷逻辑模型存在性的条件在MA假设下等价，并推导相关基数性质，还涉及Borel矩形与模型论问题。
In [Sh893], Shelah proves that (on a stationary set of cardinals) an AEC has not too many models or every model has extensions of arbitrary cardinality. We show that, if we assume limited amalgamation, then the second condition holds for a larger class of cardinals.
A subset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $μ$ and a Borel set $B$ containing $X$ such that $μ(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric space $K$, a continuous function $f : K \to G$ and a Borel set $B$ containing $X$ such that $f^{-1}(gBh)$ is meager in $K$ for every $g,h \in G$. We calculate (in $ZFC$) the four cardinal invariants ($\rm add$, $\rm cov$, $\rm non$, $\rm cof$) of these two $σ$-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb{Z}…
Generalized Descriptive Set Theory at Singular Cardinals of Countable Cofinality
本文在可数余数的奇异基数上发展了描述集合论，研究了λ-波兰空间、λ-博雷尔空间及相关集合的性质，并推广了经典完美集性质。
Descriptive set theory and forcing; How to prove theorems about Borel sets the hard way
这些是我1993年春季学期在威斯康星大学所讲的一门课的讲义。第1部分关注Borel层次结构。第13节包含一个未发表的定理Fremlin关于Borel层次和MA。第14节和第15节包含了关于Cohen和随机实模型中Borel层次结构长度的新结果。第2部分包含标准结果的理论分析集。第25节包含Harrington定理，即任意基数的$Π^1_2$集是相容的。第3部分是通常的分离定理。第四部分给出了Gandy强迫的一些应用。我们扭转了通常的趋势，并使用强制参数，而不是Baire类别。特别地，关于$Π^0_α$ hyp-sets的Louveau定理有一个更简单的使用强迫的证明。
In this work, we continue the tradition initiated by Geschke, 2011 of viewing the uncountable Borel chromatic number of analytic graphs as cardinal invariants of the continuum. We show that various uncountable Borel chromatic numbers of closed graphs can be consistently different, as well as consistently equal to the continuum. This is done using arguments that are typical to Axiom A forcing notions.
Given an analytic equivalence relation, we tend to wonder whether it is Borel. When it is non Borel, there is always the hope it will be Borel on a "large" set -- nonmeager or of positive measure. That has led Kanovei, Sabok and Zapletal to ask whether every proper $σ$ ideal satisfies the following property: given $E$ an analytic equivalence relation with Borel classes, there exists a set $B$ which is Borel and $I$-positive such that $E\restriction_{B}$ is Borel. We propose a related problem -- does every proper $σ$ ideal satisfy: given $A$ an analytic subset of the plane with Borel sections, …
We are interested in examples of a.e.c. with amalgamation having some (extreme) behaviour concerning types. Note we deal with k being sequence-local, i.e. local for increasing chains of length a regular cardinal (for types, equality of all restrictions imply equality). . For any cardinal theta>= aleph_0 we construct an a.e.c. with amalgamation k with L.S.T.(k) = theta, |tau_K| = theta such that {kappa : kappa is a regular cardinal and K is not (2^kappa, kappa)-sequence-local} is maximal. In fact we have a direct characterization of this class of cardinals: the regular kappa such that there is …
A space with only Borel subsets
米克洛什Laczkovich问是否存在一个Haussdorff（甚至正常）空间中的每一个子集是博雷尔但它不是微薄的。最后一个条件的动机是，在MA_kappa下，基数kappa的实数的每个子空间都具有所有子集都是F_sigma的性质，然而马丁公理也暗示这些子集是贫乏的。在这里，我们回答Laczkovich的问题。
Orthogonal Families of Real Sequences
对于x和y序列的实数定义内积（x，y）= x（0）y（0）+ x（1）y（1）+.它可能不是有限的，甚至不存在。我们说x和y正交当且仅当（x，y）收敛且等于0。定义l_p为所有实数序列x的集合，使得|（0）|^p +|（1）|^p +..收敛。对于希尔伯特空间，l_2，任何两两正交序列族必可数。Thm 1.存在一个尺寸连续的两两正交族F，使得F是l_p的子集，其中p>2。已知实序列中存在一族由两两正交元组成的连续统。Thm 2.实序列存在完美极大正交元素族。Abian提出的问题是什么是可能的基数的最大正交家庭。Thm 3.在Cohen实模型中存在极大正交集基数ω 1，但不存在基数k满足ω 1 < k < c的极大正交集. Thm 4.对ZFC的任意可数标准模型M和M中的基数k使得M满足k^ω=k，存在ccc泛型扩张M[G]使得M[G]的连续统为k，且在M[G]中对任意小于或等于k的无穷基数i存在极大正交基数i族. Thm 5. (MA_k（σ-centered））设X的基数小于或等于k，X只包含l_2中的1/2个元素，且对X中的每个不同对x，y，内积（x，y）收敛。则存在一个z使得z与X的每个元素正交。Thm 6. (a)存在l_2中的极大正交族X，使得对所有n，1 \leq n \leq ω，存在基数n的Y，（X并Y）是$\rr^\om$中的极大正交族.此外，每个包含$X$的极大…
Cardinal Characteristics and Computability
本文将连续统基数特征引入可计算性理论，研究预言机性质的层级，详细证明支配函数与极大独立族的Medvedev等价性，并将构造扩展至$ω$-可计算逼近集与极大理想独立族。
We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [8] holds while the classical Borel Conjecture fails.
Maximal Prikry Sequences
本文研究了在无内模型包含Woodin基数的情况下，Jensen-Steel核心模型K的覆盖机制，并证明了在某些条件下存在极大Prikry序列，且该结果是最佳的。
Cardinal invariants of idealized Miller null sets
本文研究了理想化Miller空集的基数不变量，通过计算典型Borel理想参数化的$σ$-理想，扩展了Cichoń的最大值。
In this work we study the uncountable Borel chromatic numbers, defined by Geschke (2011) as cardinal characteristics of the continuum, of low complexity graphs. We show that a strong form of locally countable graphs with compact totally disconnected set of vertices have Borel chromatic number bounded by the continuum of the ground model in the model obtained by adding $\aleph_2$ Laver reals. From this, we answer a question from Geschke and the second author (2022), and another question from Fisher, Friedman and Khomskii (2014) concerning regularity properties of subsets of the real line.
We introduce the notion of a `pure` Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If $(λ_i : i \le α<\aleph_1)$ is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$(<λ_0)$, there is an $L_{ω_1,ω}$ -sentence $ψ$ whose models form a pure AEC and (1) The models of $ψ$ satisfy JEP$(<λ_0)$, while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist $2^{λ_i^+}$ non-isomorphic maximal mod…
Completely nonmeasurable unions
假设不存在小于$2^ω$的准可测基数。（$κ$是拟可测的，如果存在$κ$-$κ$的子集的加性理想$\ci $，使得布尔代数$P(κ)/\ci$满足c.c.c。我们发现对于c.c.c. $σ$ -不可数波兰空间的子集的Borel基的理想I，如果$\cal A$是来自I的子集的点有限族，则存在$\cal A$的两两不相交的子族的不可数集合，其并集是完全不可测的，即其与每个非小博雷尔集的交集不属于由Borel集和理想I生成的$σ$ -域。这个结果是四极定理的推广。
Maximal Eventually Different Families of Computable Functions
研究可计算函数的最大最终不同（MED）族的质量问题，通过定义MED族质量问题并与Lempp等人定义的比较，借助Greenberg等人的框架，展示Schrittesser构建的有效封闭MED族，得出相关结果。
Partitioning the real line into Borel sets
本文研究实数线可被划分为多少个 Borel 集或闭集，证明除 Hausdorff 的 $\aleph_1$ 结果外，可能基数的谱在力迫扩张中可相当任意。
In this note we derive a property of maximal ideal-independent subsets of boolean algebras which has corollaries regarding the continuum cardinals p and s_mm(P(omega)/fin).
We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on $\mathbb{R}$. Notably, we explore several structural properties of Borel subsets of $\mathbb{R}$ that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes …
A family $\mathscr{I} \subseteq [ω]^ω$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal I$ and $A \in \mathscr{I} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i < n} X_i$ is infinite, is said to be ideal independent. An ideal independent family which is maximal under inclusion is said to be a maximal ideal independent family and the least cardinality of such family is denoted $\mathfrak{s}_{mm}$. We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, which in particular establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of un…
We study uniqueness of limit models in abstract elementary classes (AECs) with no maximal models. We prove (assuming instances of diamonds) that categoricity in a cardinal of the form $μ^{+(n + 1)}$ implies the uniqueness of limit models of cardinality $μ^{+}, μ^{++}, \ldots, μ^{+n}$. This sheds light on a paper of Shelah and Villaveces, who were the first to consider uniqueness of limit models in this context. We also prove that (again assuming instances of diamonds) in an AEC with no maximal models, tameness (a locality property for types) together with categoricity in a proper class of card…
We investigate when a Borel graph admits a (Borel or measurable) orientation with outdegree bounded by $k$ for various cardinals $k$. We show that for a p.m.p. graph $G$, a measurable orientation can be found when $k$ is larger than the normalized cost of the restriction of $G$ to any positive measure subset. Using an idea of Conley and Tamuz, we can also find Borel orientations of graphs with subexponential growth; however, for every $k$ we also find graphs which admit measurable orientations with outdegree bounded by $k$ but no such Borel orientations. Finally, for special values of $k$ we b…
On Borel sets in ideal topologies
本文研究了理想拓扑中Borel集的性质，证明了Borel层次不崩溃，且在特定条件下Borel集与分析集一致，并验证了近似引理在理想拓扑下的成立。
Theorem: There is a {\em complete sentence} $ϕ$ of $L_{ω_1,ω}$ such that $ϕ$ has maximal models in a set of cardinals $λ$ that is cofinal in the first measurable $μ$ while $ϕ$ has no maximal models in any $χ\geq μ$.
This is the Ph.D. thesis of the author, which was written under the supervision of Michael Hrušák at UNAM. The main contributions of this thesis are the following: There is a $+$-Ramsey \textsf{MAD} family. This answers an old question of Michael Hrušák. There are no $P$-points in the Silver model, answering a question of Michael Hrušák (this is joint work with David Chodounský. The statement \textquotedblleft There are no $P$-points\textquotedblright\ is consistent with the continuum being arbitrarily large, this answers an open question regarding $P$-points. Every Miller indestructible \text…
Let $\mathfrak{i}$ denote the minimal cardinality of a maximal independent family and let $\mathfrak{a}_T$ denote the minimal cardinality of a maximal family of pairwise almost disjoint subtrees of $2^{<ω}$. Using a countable support iteration of proper, $^ωω$-bounding posets of length $ω_2$ over a model of CH, we show that consistently $\mathfrak{i}<\mathfrak{a}_T$. Moreover, we show that the inequality can be witnessed by a co-analytic maximal independent family of size $\aleph_1$ in the presence of a $Δ^1_3$ definable well-order of the reals. The main result of the paper can be viewed as a …
On maximal ladders
本文研究了最大n-阶梯的性质，证明在Add(ω,ω_ω)下，最大n-阶梯的基数为ℵ_{n-1}，并探讨了相关问题的致密性。
Clones on regular cardinals
研究了无限集X上克隆格的结构。我们首先观察到超滤子自然地诱导克隆;这产生了罗森博格定理的一个简单证明：“在一个大小为kappa的集合上有2^2^kappa多个最大（=预完全）克隆。“我们在这里构造的克隆并不包含所有的一元函数。然后，我们研究克隆，包含所有的一元函数。利用一个强负划分定理，我们证明了对于许多基数kappa，在一个大小为kappa的集合上有2^2^kappa许多这样的克隆。最后，我们证明了在弱紧基数上恰好存在2个包含所有一元函数的极大克隆。
Classes de Wadge potentielles et théorèmes d'uniformisation partielle
我们想给一个建设尽可能简单的一个Borel子集的产品两个波兰空间。这引入了潜在Wadge类的概念。除此之外，我们研究了非潜在的闭集，通过证明Hurewicz的结果。这导致部分一致化定理，在大集，在基数或Baire范畴的意义。
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