In this paper we are interested in finding and evaluating cardinal characteristics of the continuum that appear in large-scale topology, usually as the smallest weights of coarse structures that belong to certain classes (indiscrete, inseparated, large) of finitary or locally finite coarse structures on $ω$. Besides well-known cardinals $\mathfrak b,\mathfrak d,\mathfrak c$ we shall encounter two new cardinals $\mathsf Δ$ and $\mathsf Σ$, defined as the smallest weight of a finitary coarse structure on $ω$ which contains no discrete subspaces and no asymptotically separated sets, respectively. We prove that $\max\{\mathfrak b,\mathfrak s,\mathrm{cov}(\mathcal N)\}\le\mathsf Δ\le\mathsf Σ\le\mathrm{non}(\mathcal M)$, but we do not know if the cardinals $\mathsf Δ,\mathsf Σ,\mathrm{non}(\mathcal M)$ can be separated in suitable models of ZFC.
We prove that consistently there is a singular cardinal $κ$ of uncountable cofinality such that $2^κ$ is weakly inaccessible, and every regular cardinal strictly between $κ$ and $2^κ$ is the character of some uniform ultrafilter on $κ$.
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.
Assuming the existence of a strong cardinal, we find a model of ZFC in which for each uncountable regular cardinal $λ,$ there is no universal graph of size $λ$.
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 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 $κ^{++}$.
Assuming the existence of a supercompact cardinal and an inaccessible above it, we construct a model of ZFC, in which all uncountable regular cardinals are inaccessible in HOD.
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…
We point out that a construction by Butkovičová of a chain of length $\mathfrak{c}^+$ in the Rudin-Frolík order on $βω$ can easily be adapted to produce, given an uncountable cardinal $κ$, a chain of length $(2^κ)^+$ in the Rudin-Frolík order on $βκ$.
The {\em Singular Cardinal Hypothesis} (SCH) is one of the most classical combinatorial principles in set theory. It says that if $κ$ is singular strong limit, then $2^κ=κ^+$. We prove that given a singular cardinal $κ$ of {\em cofinality} $η$ in the ground model, which is a limit of suitable large cardinals, and $η^+=\aleph_γ$, then there is a forcing extension which preserves cardinals and cofinalities up to and including $η$, such that $κ$ becomes $\aleph_{γ+η}$, and SCH fails at $κ$. Furthermore, if $η$ is not an $\aleph$-fixed point, then in our model, SCH fails at $\aleph_η$. Our large c…
Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $\mathfrak p(κ)=\mathfrak t(κ)$ or there is a $(\mathfrak p(κ),λ)$-gap of club-supported slaloms for some $λ< \mathfrak p(κ)$. While the existence of such gaps is unclear, this is a promising step to lift Malliaris and Shelah's proof of $\mathfrak p=\mathfrak t$ to uncountable cardinals. We do analyze gaps of slaloms and, in particular, show that $\mathfrak p(…
We pursue the idea of generalizing Hindman's Theorem to uncountable cardinalities, by analogy with the way in which Ramsey's Theorem can be generalized to weakly compact cardinals. But unlike Ramsey's Theorem, the outcome of this paper is that the natural generalizations of Hindman's Theorem proposed here tend to fail at all uncountable cardinals.
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$λ$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq λ$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
We establish some similarities/analogies between uncountable cardinals or powersets and the class $V$ of all sets. They concern mainly the Boolean algebras ${\cal P}(κ)$, for a regular cardinal $κ$, and ${\cal C}(V)$ (the class of subclasses of the universe $V$), endowed with some ideals, especially the ideal $[κ]^{<κ}$ for ${\cal P}(κ)$, and the ideal of sets $V$ for ${\cal C}(V)$.
We consider the following dichotomy for $Σ^0_2$ finitary relations $R$ on analytic subsets of the generalized Baire space for $κ$: either all $R$-independent sets are of size at most $κ$, or there is a $κ$-perfect $R$-independent set. This dichotomy is the uncountable version of a result found in (W. Kubiś, Proc. Amer. Math. Soc. 131 (2003), no 2.:619--623) and in (S. Shelah, Fund. Math. 159 (1999), no. 1:1--50). We prove that the above statement holds assuming $\Diamond_κ$ and the set theoretical hypothesis $I^-(κ)$, which is the modification of the hypothesis $I(κ)$ suitable for limit cardin…
We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $κ$ with the property that the collection of all initial segments of the wellordering is definable by a $Σ_1$-formula with parameter $κ$. A short argument shows that the existence of a measurable cardinal $δ$ implies that such wellorderings do not exist at $δ$-inaccessible cardinals of cofinality not equal to $δ$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a…
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…
We prove that assuming suitable cardinal arithmetic, if B is a Boolean algebra every homomorphic image of which is isomorphic to a factor, then B has locally small density. We also prove that for an (infinite) Boolean algebra B, the number of subalgebras is not smaller than the number of endomorphisms, and other related inequalities. Lastly we deal with the obtainment of the supremum of the cardinalities of sets of pairwise incomparable elements of a Boolean algebra.
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many stationary sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some stationary set, does hold for s…
Given an uncountable regular cardinal $κ$, a partial order is $κ$-stationarily layered if the collection of regular suborders of $\mathbb{P}$ of cardinality less than $κ$ is stationary in $\mathcal{P}_κ(\mathbb{P})$. We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal $κ$ is weakly compact if and only if every partial order satisfying the $κ$-chain condition is $κ$-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all $κ$-Knaster partial…
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…
We provide analogues of the results from [FMR11, CMMR13] in the reference list (which correspond to the case $κ= ω$) for arbitrary $κ$-Souslin quasi-orders on any Polish space, for $κ$ an infinite cardinal smaller than the cardinality of $\mathbb{R}$. These generalizations yield a variety of results concerning the complexity of the embeddability relation between graphs or lattices of size $κ$, the isometric embeddability relation between complete metric spaces of density character $κ$, and the linear isometric embeddability relation between (real or complex) Banach spaces of density $κ$.
In this paper we investigate some properties of forcing which can be considered "nice" in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular cardinal, cannot be too nice and must cause some "damage" to the structure of cardinals and stationary sets. As a consequence there is no analogue to the Prikry forcing, in terms of "nice" properties, when changing cofinalities to be uncountable.
By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space $X$ is covered by a $K_σ$ subset of $X$ if and only if it does not contain a closed-in-$X$ subset homeomorphic to the Baire space ${}^ωω$. We consider the analogous statement (which we call Hurewicz dichotomy) for $Σ^1_1$ subsets of the generalized Baire space ${}^κκ$ for a given uncountable cardinal $κ$ with $κ=κ^{<κ}$, and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the Generalized Continuum Hypothesis (GCH) hol…
We discuss the rainbow Ramsey theorems at limit cardinals and successors of singular cardinals, addressing some questions in \cite{MR2354904} and \cite{MR2902230}. In particular, we show for inaccessible $κ$, $κ\to^{poly}(κ)^2_{2-bdd}$ does not characterize weak compactness and for singular $κ$, $\mathrm{GCH}+\square_κ$ implies $κ^+\not\to^{poly} (η)^2_{<κ-bdd}$ for any $η\geq cf(κ)^+$ and $κ^+\to^{poly} (ν)^2_{<κ-bdd}$ for any $ν<cf(κ)^+$. We also provide a simplified construction of a model for $ω_2\not\to^{poly} (ω_1)^2_{2-bdd}$ originally constructed in \cite{MR2902230} and show the witnes…
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.
Let kappa be a regular uncountable cardinal and lambda >=kappa^+ . The principle of stationary reflection for P_kappa lambda has been successful in settling problems of infinite combinatorics in the case kappa=omega_1. For a greater kappa the principle is known to fail at some lambda. This note shows that it fails at every lambda if kappa is the successor of a regular uncountable cardinal or kappa is countably closed.
We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak{s}_θ,\mathfrak{p}_θ,\mathfrak{g}_θ,\mathfrak{r}_θ,\mathfrak{t}_θ$ at uncountable regular cardinals $θ$. Motivated by a theorem of Raghavan-Shelah who proved that $\mathfrak{s}_θ\leq\mathfrak{b}_θ$, we explore in the first part of the paper the consistency of inequalities comparing $\mathfrak{s}_θ$ with $\mathfrak{p}_θ$ and $\mathfrak{g}_θ$. In the second part of the paper we study variations of the extender-based Radin forcing to establish several consistency results c…
Answering some of the main questions from [MR13], we show that whenever $κ$ is a cardinal satisfying $κ^{< κ} = κ> ω$, then the embeddability relation between $κ$-sized structures is strongly invariantly universal, and hence complete for ($κ$-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [LR05,FMR11,Wil14,CMR17].
We extend a dichotomy between 1-basedness and supersimplicity proved in a previous paper. The generalization we get is to arbitrary language, with no restrictions on the topology (we do not demand type-definabilty of the open set in the definition of essential 1-basedness). We conclude that every (possibly uncountable) hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the language is countable.
Strong Completeness of Provability Logic for Uncountable Languages
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所数学学院）；Institut für diskrete Mathematik und Geometrie, TU Wien（维也纳工业大学离散数学与几何研究所）
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}.
Shelah has provided sufficient conditions for an $L_{ω_1, ω}$-sentence $ψ$ to have arbitrarily large models and for a Morley-like theorem to hold of $ψ$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $ψ$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $ψ$ be a universal $L_{ω_{1}, ω}$-sentence. - If $ψ$ is categorical in $\aleph_{0}$ and $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, …
Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V. In particular, a model with all regular uncountable cardinals measurable in HOD is constructed.
Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal $κ$, there are no $Σ^1_1(κ)-κ-$mad families.
We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and forcing axioms imply that there is a homogeneous closed unbounded subset of $ω_1$ for every colouring of the finite sets of countable ordinals that is definable by a $Σ_1$-formula that only uses the cardinal $ω_1$ and real numbers as parameters. Moreover, it is shown that certain large cardinal properties cause analogous partition properties to hold at the given …
Assume G.C.H. and kappa is the first uncountable cardinal such that there is a kappa-free abelian group which is not a Whitehead (abelian) group. We prove that kappa is necessarily an inaccessible cardinal
In this note we collect some known information and prove new results about the small uncountable cardinal $\mathfrak q_0$. The cardinal $\mathfrak q_0$ is defined as the smallest cardinality $|A|$ of a subset $A\subset \mathbb R$ which is not a $Q$-set (a subspace $A\subset\mathbb R$ is called a $Q$-set if each subset $B\subset A$ is of type $F_σ$ in $A$). We present a simple proof of a folklore fact that $\mathfrak p\le\mathfrak q_0\le\min\{\mathfrak b,\mathrm{non}(\mathcal N),\log(\mathfrak c^+)\}$, and also establish the consistency of a number of strict inequalities between the cardinal $\…
Assume $\mathsf{ZF}+\mathsf{AD}+V=L(\mathbb{R})$ and let $κ<Θ$ be an uncountable cardinal. We show that $κ$ is Jónsson, and that if $\mathrm{cof}(κ)=ω$ then $κ$ is Rowbottom. We also establish some other partition properties.
For an uncountable regular cardinal κwe let \nabla_κ(A) be the statement that A \subset κand for all regular θ> κ, the set of all X \in [θ]^<κsuch that X \cap κ\in κand otp(X \cap OR) is a cardinal in L[A \cap X \cap κ] is stationary. We had shown earlier that \nabla_{ω_1}(A) can hold in a generic extension of L. We now prove that \nabla_{ω_2}(A) can hold in a semi-proper generic extension of L, whereas \nabla_{ω_3}(0) is equivalent with the existence of 0^#.
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…
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.
Let kappa be an uncountable cardinal and the edges of a complete graph with kappa vertices be colored with aleph_0 colors. For kappa >2^{aleph_0} the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if kappa <= 2^{aleph_0}, then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when kappa <= 2^{aleph_0}. In particular, we are concerned with the case aleph_1 <= kappa <= aleph_omega
Let kappa be a regular uncountable cardinal and lambda > kappa a singular strong limit cardinal. We give a new characterization of the nonstationary subsets of P_kappa (lambda) and use this to prove that the nonstationary ideal on P_kappa (lambda) is nowhere precipitous.
Covering matrices were introduced by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of his work and in subsequent work with Sharon, he isolated two reflection principles, $\mathrm{CP}$ and $\mathrm{S}$, which may hold of covering matrices. In this paper, we continue previous work of the author investigating connections between failures of $\mathrm{CP}$ and $\mathrm{S}$ and variations on Jensen's square principle. We prove that, for a regular cardinal $λ> ω_1$, assuming large cardinals, $\square(λ, 2)$ is consistent with $\mathrm{C…
A subset $\mathcal X$ of a C*-algebra $\mathcal A$ is called irredundant if no $A\in \mathcal X$ belongs to the C*-subalgebra of $\mathcal A$ generated by $\mathcal X\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum. There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic ax…
