Martin Goldstern, Saharon Shelah
For g < f in omega^omega we define c(f,g) be the least number of uniform trees with g-splitting needed to cover a uniform tree with f-splitting. We show that we can simultaneously force aleph_1 many different values for different functions (f,g). In the language of Blass: There may be aleph_1 many distinct uniform Pi^0_1 characteristics.
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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.
We prove some consistency results about b(lambda) and d(lambda), which are natural generalisations of the cardinal invariants of the continuum b and d. We also define invariants b_cl(lambda) and d_cl(lambda), and prove that almost always b(lambda) = b_cl(lambda) and d(lambda)=d_cl(lambda)
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 $\aleph_2$.
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.
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.
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。
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…
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。
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…
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…
This is a revision (and partial retraction) of my previous abstarct. Let $λ(X)$ denote Lebesgue measure. If $X\subseteq [0,1]$ and $r \in (0,1)$ then the $r$-Hausdorff capacity of $X$ is denoted by $H^r(X)$ and is defined to be the infimum of all $\sum_{i=0}^\infty λ(I_i)^r$ where $\{I_i\}_{i\inω}$ is a cover of $X$ by intervals. The $r$ Hausdorff capacity has the same null sets as the $r$-Hausdorff measure which is familiar from the theory of fractal dimension. It is shown that, given $r < 1$, it is possible to enlarge a model of set theory, $V$, by a generic extension $V[G]$ so that the real…
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 $Sym(κ)$.
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…
A function f from reals to reals (f:R->R) is almost continuous (in the sense of Stallings) iff every open set in the plane which contains the graph of f contains the graph of a continuous function. Natkaniec showed that for any family F of continuum many real functions there exists g:R->R such that f+g is almost continuous for every f in F. Let AA be the smallest cardinality of a family F of real functions for which there is no g:R->R with the property that f+g is almost continuous for every f in F. Thus Natkaniec showed that AA is strictly greater than the continuum. He asked if anything more…
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。
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。
A survey of recent results concerning cardinal invariants of measure and category. Submitted as a chapter of the upcoming Handbook of Set Theory.
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…
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}…
We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (boldface Sigma^0_2 and, under suitable restrictions, Pi^0_2) are shown to have pleasant properties, related to Baire category. We construct models of set theory where (unrestricted) boldface Pi^0_2-characteristics behave quite chaotically and no new characteristics appear at higher complexity levels. We also discuss some characteristics associated with partition theorems and we present, in an appendix, a simplified p…
Whenever x is a tame cardinal invariant and ZFC+large cardinals proves that x=aleph one implies WCG then ZFC+large cardinals proves that x=aleph one implies b=aleph one, and b=aleph one implies WCG. Here WCG is a certain prediction principle on omega one. This theorem is one of the many possible interpolation theorems of this kind.
We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact satura…
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.
Constant prediction and evasion number, I: Generalization and variants
本文将常数逃避概念应用于二元关系，建立了源自已知基数不变量的常数逃避数和常数预测数等新不变量，并给出了相关的极限与一致性结果。
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。
Transfinite Schreier families and cardinal invariants
该研究探讨超限Schreier层级对可数极限序数基本序列选择的依赖，关联区间端点函数，证明边界数、控制数与基本序列选择的等价条件，结合相关定理与吸收构造得出结论。
For every regular cardinal kappa there exists a simple complete Boolean algebra with kappa generators.
We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B_0 times B_1)= max(irr(B_0),irr(B_1)). We prove consistency of the statement ``there is a Boolean algebra B such that irr(B)<s(B otimes B)'' and we force a superatomic Boolean algebra B_* such that s(B_*)=inc(B_*)=kappa, irr(B_*)=Id(B_*)=kappa^+ and Sub(B_*)=2^(kappa^+). Next we force a superatomic algebra B_0 such that irr(B_0)<inc(B_0) and a superatomic algebra B_1 such that t(B_1)>Aut(B_1). Finally we show that …
For a ring R, denote by Spec^R_kappa(Gamma) the kappa-spectrum of the Gamma-invariant of strongly uniform right R-modules. Recent realization techniques of Goodearl and Wehrung show that Spec^R_{aleph_1}(Gamma) is full for suitable von Neumann regular algebras R, but the techniques do not extend to cardinals kappa>aleph_1. By a direct construction, we prove that for any field F and any regular uncountable cardinal kappa there is an F-algebra R such that Spec^R_kappa(Gamma) is full. We also derive some consequences for the complexity of Ziegler spectra of infinite dimensional algebras.
Game-theoretic variants of splitting number
本文通过引入三种博弈论视角下的分裂数变体，探讨了分裂数与无限博弈结合后的新基数不变量，并证明了其中某些数与连续体大小或分裂数的关系。
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an posi…
The structure of quotient Boolean algebras in terms of cardinal invariants is investigated. Some results of Gitik and Shelah regarding atomless ideals are reproved and proofs are significantly simplified.
Relative cofinality of ideals
该研究引入理想共尾性不变量的双参数修正版本，纳入理想对相互作用，确定ω上关键理想对的不变量值，二分实数线理想对并研究极大理想的相对共尾性，推进基类结构研究。
Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise different. In addition we show that, in the same forcing extension, for two other types of simple cardinal characteristics parametrised by reals (localisation and anti-localisation cardinals), for uncountably many parameters the corresponding cardinals are pairwise different.
We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n -independent. The properties of these classes (n-free and omega-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, n Ind, which is the supremum of the cardinalities of n-independent subsets; i_n, the minimum size of a maximal n -i…
Bi-invariant types, reliably invariant types, and the comb tree property
该研究引入双不变型、可靠不变型等几类特殊不变型，建立其与梳树性质、反链树性质等模型论树性质的关联，推广了相关未发表结果，刻画了多类理论的局部特征与Kim-分叉、划分的性质。
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