Francis Adams, Jindrich Zapletal
We study several cardinal characteristics of closed graphs G on compact metrizable spaces. In particular, we address the question when it is consistent for the bounding number to be strictly smaller than the smallest size of a set not covered by countably many compact G-anticliques. We also provide a descriptive set theoretic characterization of the class of analytic graphs with countable coloring number.
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There is an optimal way to increase certain cardinal invariants of the continuum.
This paper studies cardinal invariants b_kappa and t_kappa, the natural generalizations of the invariants b and t to a regular cardinal kappa.
We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncountable regular cardinal $κ$, $\mathfrak{b}(κ) = κ^{+}$ implies $\mathfrak{a}(κ) = κ^{+}$. This improves an earlier result of Blass, Hyttinen, and Zhang. It is also shown that if $κ\geq {\beth}_ω$ is an uncountable regular cardinal, then $\mathfrak{d}(κ) \leq \mathfrak{r}(κ)$. This result partially dualizes an earlier theorem of the authors.
Cardinal invariants associated with Hausdorff measures
本文研究了通过Mathiasforcing分离豪斯多夫测度0理想和勒贝格空理想基数不变量的方法。
The present status of the problems in my book "Cardinal Invariants on Boolean algebras" (Birkhauser 1996) is described, with a description of solutions or partial solutions, and references.
Using GCH, we force the following: There are continuum many simple cardinal characteristics with pairwise different values.
We prove that the strong polarized relation of $θ$ above $ω$ applied simultaneously for every cardinal in the interval $[\aleph_1,\aleph]$ is consistent. We conclude that this positive relation is consistent for every cardinal invariant on the continuum. We show that similar results hold for a supercompact cardinal, and for a strong limit singular under some assumptions.
Cardinal invariants above the continuum
我们证明了一些关于b（lambda）和d（lambda）的相容性结果，这些结果是连续统b和d的基数不变量的自然推广。我们还定义了不变量b_cl（lambda）和d_cl（lambda），并证明了几乎总是b（lambda）= b_cl（lambda）和d（lambda）=d_cl（lambda）
A cardinal kappa is countably closed if mu^omega < kappa whenever mu < kappa. Assume that there is no inner model with a Woodin cardinal and that every set has a sharp. Let K be the core model. Assume that kappa is a countably closed cardinal and that alpha is a successor cardinal of K with kappa < alpha < kappa^+. Then cf( alpha ) = kappa. In particular, K computes successors of countably closed singular cardinals correctly. (The hypothesis of countable closure is not required; see "Weak covering without countable closure", W. J. Mitchell and E. Schimmerling, Math. Res. Lett., Vol. 2, No. 5, …
We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $ω$ of asymptotic density $0$. We obtain an upper bound on the $\ast$-covering number, sometimes also called the weak covering number, of this ideal by proving in Section \ref{sec:covz0} that ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0}) \leq \mathfrak{d}$. In Section \ref{sec:skbk} we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We pro…
Let "ex" be the cardinality of the smallest independent family of subsets of omega (independent means that all nontrivial Boolean combinations are infinite) which cannot be extended to a homogeneous independent family. "Homogeneous" means that every finite partial map (points to points, sets to sets) that is compatible with the family is induced by a permutation of omega. Despite its apparently ``complex'' (second order) definition, this cardinal is equal to a well-known ``simple'' cardinal invariant from Cichon's diagram.
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)…
We extend the applications of the techniques used in Arch Math Logic 52:261-278, 2013, to present various examples of consistency results where some cardinal invariants of the continuum take arbitrary regular values with the size of the continuum being bigger than $\aleph_2$.
The Hausdorff measure due to Davies and Rogers and its cardinal invariants
本文研究了Davies和Rogers构造的Hausdorff测度的基数不变量特性。
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…
In [6], given a metrizable profinite group $G$, a cardinal invariant of the continuum $\mathfrak{fm}(G)$ was introduced, and a positive solution to the Haar Measure Problem for $G$ was given under the assumption that $\mathrm{non}(\mathcal{N}) \leq \mathfrak{fm}(G)$. We prove here that it is consistent with ZFC that there is a metrizable profinite group $G_*$ such that $\mathrm{non}(\mathcal{N}) > \mathfrak{fm}(G_*)$, thus demonstrating that the strategy of [6] does not suffice for a general solution to the Haar Measure Problem.
Many simple cardinal invariants
对于ω ^ω中的g < f，我们定义c（f，g）为覆盖一个具有f-分裂的均匀树所需的具有g-分裂的均匀树的最少数量。我们表明，我们可以同时迫使aleph_1许多不同的值为不同的功能（f，g）。用布拉斯的话来说：可能有许多明显的一致的Pi ^[0_1]特征。
Game-theoretic variants of cardinal invariants
本文研究了连续统基数不变量的博弈论变体，分析了reaping、bounding、dominating数及null理想加法数的博弈性质，并确定了玩家的获胜策略。
We deal with some of problems posed by Monk and related to cardinal invariant of ultraproducts of Boolean algebras. We also introduce and investigate some new cardinal invariants.
The additivity spectrum ADD(I) of an ideal I is the set of all regular cardinals kappa such that there is an increasing chain {A_alpha:alpha<kappa\} in the ideal I such that the union of the chain is not in I. We investigate which set A of regular cardinals can be the additivity spectrum of certain ideals. Assume that I=B or I=N, where B denotes the sigma-ideal generated by the compact subsets of the Baire space omega^omega, and N is the ideal of the null sets. For countable sets we give a full characterization of the additivity spectrum of I: a non-empty countable set A of uncountable regular…
Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal $\mathfrak r$. This is achieved through analyzing nowhere reaping families of subsets of $\mathbb N$ and through applying a paving lemma which is a consequence of a p…
Cardinal invariants associated with the combinatorics of the uniformity number of the ideal of meager-additive sets
本文综述了与贫瘠可加集理想均匀数相关的基数不变量 $\mathfrak{b}_b^\mathsf{eq}$ 及其对偶 $\mathfrak{d}_b^\mathsf{eq}$，探讨它们与 Cichoń 图中基数的联系，并提出若干公开问题。
A club consisting of former regulars is added to an inaccessible cardinal, without changing cofinalities outside it. The initial assumption is optimal. A variation of the Radin forcing without a top measurable cardinal is introduced for this.
Forcing and cardinal invariants of universally null sets
本文研究普遍空集上的基数不变量，证明了多个相关不变量之间的不等式关系，并探讨了在特定假设下的一致性结果。
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。
We investigate which filters on $ω$ can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in $[ω]^ω$). We prove the following results: - Many classical examples of nice tall filters contain no towers (in ZFC). - It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well). - It is consistent that all towers generate non-meager filters, in particular (consistently) Borel filters do not contain towers. - The statement "Every ultrafilter contains towers." is independent of ZF…
Cichoń's maximum with cardinals of the closed null ideal
本文研究了闭零测集理想 $\mathcal{E}$ 的基数特征，证明其一致性与覆盖数可作为独立取值加入 Cichoń 最大值，扩展了经典基数不等式链的一致性结果。
We construct Boolean Algebras answering questions of Monk on cardinal invariants. The results are proved in ZFC (rather than giving consistency results). We deal with the existence of superatomic Boolean Algebras with ``few automorphisms'', with entangled sequences of linear orders, and with semi-ZFC examples of the non-attainment of the spread (and hL, hd).
Keisler's Theorem and Cardinal Invariants
本文研究了Keisler定理的不同变体及其与基数不变量的关系，证明了在特定条件下该定理的成立与某些基数的等价性有关。
Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Here, we study the indestructibility properties of remarkable cardinals. We show that if $κ$ is remarkable, then there is a forcing extension in which the remarkability of $κ$ becomes indestructible by all $\ltκ$-closed $\leqκ$-distributive forcing and all two-step iterations of the form ${\rm Add}(κ,θ)*\dot{\mathbb R}$, where $\dot{\mathbb R}$ is forced to be $\ltκ$-closed and $\leqκ$-di…
Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $κ=λ^+$ for some $λ\geq ω$ and $\mathfrak b(κ)=κ^+$ then $\mathfrak a_e(κ)=\mathfrak a_p(κ)=κ^+$. If, additionally, $2^{<λ}=λ$ then $\mathfrak a_g(κ)=κ^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak d(κ)$ in terms of $\mathfrak r(κ)$, including $\mathfrak d(κ)\leq \mathfrak r_σ(κ)\leq \mathrm{cof}([\mathfrak r(κ)]^ω)$, and…
The Higher Closed Null Ideal(s)
本文将闭勒贝格零理想推广到更高康托空间，引入基于slalom的候选理想并研究其组合性质及基数不变量关系。
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…
Cardinal Invariants Associated with Hausdorff Capacities
这是对我以前的论文的修改（和部分撤回）。设$λ(X)$表示Lebesgue测度。如果$X\subseteq [0,1]$和$r \in (0,1)$，则$X$的$r$-Hausdorff容量由$H^r(X)$表示，并且被定义为所有$\sum_{i=0}^\infty λ(I_i)^r$的下确界，其中$\{I_i\}_{i\inω}$是$X$的区间覆盖。$r$ Hausdorff容量与分形维数理论中的$r$ -Hausdorff测度具有相同的空集.它表明，给定$r < 1$，它是可能的集合论模型，$V$，通过一般的扩展$V[G]$，使现实的$V$有勒贝格测度为零，但仍然有积极的$r$ -Hausdorff容量。
Limit Groups and Automorphisms of $κ$-Existentially Closed Groups
本文研究κ-存在闭群的自同构群基数问题，证明了基数为κ的κ-存在闭群的自同构群基数为2^κ，并回答了Kegel关于任意不可数基数κ上存在泛群的问题。
We prove that for infinite cardinals $κ<λ$ the alternating group $Alt(λ)$ (of even permutations) of $λ$ is not embeddable into the symmetric group $Sym(κ)$ (of all permutations) of $κ$. To prove this fact we introduce and study several monotone cardinal group invariants which take value $κ$ on the groups $Alt(κ)$ and $Sym(κ)$.
Let $\mathcal{E}$ be the $σ$-ideal generated by the closed measure zero sets of reals. We use an ultrafilter-extendable matrix iteration of ccc posets to force that, for $\mathcal{E}$, their associated cardinal characteristics (i.e.\ additivity, covering, uniformity and cofinality) are pairwise different.
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 $E_0$.
Given an ideal $I$ on $ω$ let $a(I) $ ($\bar{a}(I)$) be minimum of the cardinalities of infinite (uncountable) maximal $I$-almost disjoint subsets of $[ω]^ω$, and denote $b_I$ and$d_I$ the unbounding and dominating numbers of $(ω^ω,\le_I)$. We show that (1) $a(I)>omega$ if $I$ is a summable ideal; (2) $a(Z)=ω$ and $\bar{a}(Z)\le a$ if $Z$ is a tall density ideal, (3) $b\le \bar{a}(I)$, and $b_I=b$ and $d_I=d$, for any analytic P-ideal $I$ on $ω$. Given an analytic $P$-ideal $I$ we investigate the relationship between the Sack, the $I$-bounding, $I$-dominating and $ω^ω$-bounding properties of a…
Cardinal invariants concerning functions whose sum is almost continuous
一个从实数到实数的函数f（f：R->R）是几乎连续的（在Stallings意义下）当且仅当包含f的图的平面中的每个开集包含一个连续函数的图。Natkaniec证明了对于任何连续族F，许多实函数都存在g：R->R，使得f+g对F中的每一个f几乎连续。设AA是不存在g：R->R的实函数族F的最小基数，且f+g对F中的每一个f几乎连续.因此，Natkaniec表明，AA是严格大于的连续。他问是否还有什么可以说的。我们证明了AA的共尾性大于连续统c。此外，我们通过证明AA可以等于c^+和2^c之间的任何正则基数（2^c任意大），证明了这几乎是ZFC中关于AA所能说的一切。我们还表明，AA = AD AD AD的定义类似于AA，但类的达布功能。这解决了Maliszewski和Natkaniec的另一个问题。
We give a combinatorial characterization of countable submaximal subspaces of $2^κ$. Using a parametrized version of Mathias forcing, we prove that there exists a countable submaximal subspace of $2^{ω_1}$ whilst $\mathfrak{c}=ω_2$. Combining this with previous results, we construct a disjointly tight countable irresolvable space of weight $<\mathfrak{c}$, answering a question of Bella and Hrušák.
Cardinal invariants related to density
本文研究了基于渐近密度定义的分裂数、收割数和独立性数变体，证明了多个基数不等式的一致性，并回答了arXiv:1808.02442v3中的两个问题。
Much recent work in cardinal characteristics has focused on generalizing results about $ω$ to uncountable cardinals by studying analogues of classical cardinal characteristics on the generalized Baire and Cantor spaces $κ^κ$ and $2^κ$. In this note I look at generalizations to other function spaces, focusing particularly on the space of functions $f:ω^ω\to ω^ω$. By considering classical cardinal invariants on Baire space in this setting I derive a number of "higher dimensional" analogues of such cardinals, ultimately introducing 18 new cardinal invariants, alongside a framework that allows for…
This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] we study which cardinals are characterizable by a Scott sentence $ϕ$, in the sense that $ϕ$ characterizes $κ$, if $ϕ$ has a model of size $κ$, but no models of size $κ^+$. We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products (cf. theorems 2.3, 3.4, and corollary 3.6). We also prove that if $\aleph_α$ is characterized by a Scott sentence, at least one of $\aleph_alpha$ and $\aleph_alpha^+$ is …
On cardinal invariants related to Rosenthal families and large-scale topology
本文通过引入与Rosenthal族相关的基数不变量 $\mathfrak{ros}_Γ$ 和 $Δ_Γ$，并与连续统经典基数特征比较，部分解决了已有文献中的若干问题。
Let $κ$ be an uncountable cardinal with $κ=κ^{{<}κ}$. Given a cardinal $μ$, we equip the set ${}^κμ$ consisting of all functions from $κ$ to $μ$ with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than $κ$. We prove results that allow us to separate several classes of subsets of ${}^κκ$ that consist of continuous images of closed subsets of spaces of the form ${}^κμ$. Important examples of such results are the following: (i) there is a closed subset of ${}^κκ$ that is not a continuous image of ${}^κκ$; (ii) there is an injective continuous…
We show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P, Q$, $\langle L[P],\in ,P \rangle$ and $\langle L[Q],\in ,Q \rangle$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. The theory of such models is thus invariant under set forcing. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. One outcome is that we can chara…
A survey of recent results concerning cardinal invariants of measure and category. Submitted as a chapter of the upcoming Handbook of Set Theory.
We study the structured rainbow Ramsey theory at uncountable cardinals. When compared to the usual rainbow Ramsey theory, the variation focuses on finding a rainbow subset that not only is of a certain cardinality but also satisfies certain structural constraints, such as being stationary or closed in its supremum. In the process of dealing with cardinals greater than $ω_1$, we uncover some connections between versions of Chang's Conjectures and instances of rainbow Ramsey partition relations, addressing a question raised in \cite{zhang}.
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…
Combinatorics of translations of meager and closed measure zero sets
研究贫集与闭测度零集平移的组合性质，探讨相关Borel猜想，获得贫理想可加性新刻画，并证明若干基数不变量关系。
Noetherian type and Noetherian $π$-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the \emph{cellularity}, that is the supremum of the sizes of pairwise disjoint non-empty open sets in a topological space. Building on that analogy, we study the Noetherian $π$-type of $κ$-Suslin Lines, and we are able to determine it for every $κ$ up to the first singular cardinal. We then prove a consequence of Chang's Conjecture for $\aleph_ω$ regarding the Noetheria…
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}…
For a Polish group G let cov_G be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G. For many locally compact G this cardinal is known to be consistently larger than cov(meager) which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups cov_G=cov(meager). For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces. Most re…
On $Σ_1$-Definable Closed Unbounded Sets
该文聚焦集合论中基数$κ$的$Σ_1$-可定义闭无界集，研究其对完全闭无界滤子的逼近程度及对定义复杂度的依赖，发现逼近能力差异大且依赖底层集合论宇宙性质。
The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call exact preservation theorems, I use this new technology to perform a kind of partial Lave…
A ``k-rule" is a sequence A=((A_n,B_n):n<omega) of pairwise disjoint sets B_n, each of cardinality at most k, where A_n is a subset of B_n. A set X of natural numbers (a ``real'') follows a rule A if for infinitely many n we have that the intersection of X with B_n is exactly A_n. There are obvious cardinal invariants resulting from this definition: the least number of reals needed to follow all k-rules, s_k, and the least number of k-rules without a real following all of them, r_k. We investigate these cardinal invariants and their connection to some well-known cardinals from Cichon's diagram…
Generalized cardinal invariants for an inaccessible $κ$ with compactness at $κ^{++}$
本文证明在超紧基数与弱紧基数一致的前提下，存在不可达基数 $κ$ 使得广义塔数与超滤数满足特定不等式关系，并计算了多个基数不变量的值，同时推广到传统不变量并得到不相交平稳序列性质。
The almost disjointness numbers associated to the quotients determined by the transfinite products of the ideal of finite sets are investigated. A $\mathrm{ZFC}$ lower bound involving the minimum of the classical almost disjointness and splitting numbers is proved for these characteristics. En route, it is shown that the splitting numbers associated to these quotients are all equal to the classical splitting number. Finally, it is proved to be consistent that the almost disjointness numbers associated to these quotients are all equal to the second uncountable cardinal while the bounding number…
We consider a cardinal invariant closely related to Hindman's theorem. We prove that this cardinal invariant is small in the iterated Sacks perfect set forcing model, and that its corresponding parametrized diamond principle implies the existence of union ultrafilters. As a corollary, this establishes the existence of union ultrafilters in the iterated Sacks model of Set Theory.
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