Recent zbMATH articles in MSC 03E https://zbmath.org/atom/cc/03E 2023-09-19T14:22:37.575876Z Werkzeug The closed ordinal Ramsey number $$R^{cl}(\omega^2,3) = \omega^6$$ https://zbmath.org/1516.03013 2023-09-19T14:22:37.575876Z "Mermelstein, Omer" https://zbmath.org/authors/?q=ai:mermelstein.omer Summary: Closed ordinal Ramsey numbers are a topological variant of the classical (ordinal) Ramsey numbers. We compute the exact value of the closed ordinal Ramsey number $$R^{cl}(\omega^2,3) = \omega^6$$. Local coloring problems on smooth graphs https://zbmath.org/1516.03014 2023-09-19T14:22:37.575876Z "Bernshteyn, Anton" https://zbmath.org/authors/?q=ai:bernshteyn.anton A graph $$G$$ is Borel if $$V(G)$$ is a Borel space and $$E(G)$$ is a Borel subset of $$V(G) \times V(G)$$. A locally countable graph is smooth if it admits a Borel transversal $$T$$ (i.e. every connected component of $$G$$ contains exactly one element from $$T$$). The author investigates local colorings. He constructs a smooth locally finite Borel graph $$G$$ and a local coloring problem $$\Pi$$ such that $$G$$ has a coloring $$V(G) \rightarrow \mathbb{N}$$ that solves $$\Pi$$, but no such coloring can be Borel. Reviewer: Martin Weese (Potsdam) The rearrangement number https://zbmath.org/1516.03015 2023-09-19T14:22:37.575876Z "Blass, Andreas" https://zbmath.org/authors/?q=ai:blass.andreas-raphael "Brendle, Jörg" https://zbmath.org/authors/?q=ai:brendle.jorg "Brian, Will" https://zbmath.org/authors/?q=ai:brian.william-rea "Hamkins, Joel David" https://zbmath.org/authors/?q=ai:hamkins.joel-david "Hardy, Michael" https://zbmath.org/authors/?q=ai:hardy.michael "Larson, Paul B." https://zbmath.org/authors/?q=ai:larson.paul-b Summary: How many permutations of the natural numbers are needed so that every conditionally convergent series of real numbers can be rearranged to no longer converge to the same sum? We define the rearrangement number, a new cardinal characteristic of the continuum, as the answer to this question. We compare the rearrangement number with several natural variants, for example one obtained by requiring the rearranged series to still converge but to a new, finite limit. We also compare the rearrangement number with several well-studied cardinal characteristics of the continuum. We present some new forcing constructions designed to add permutations that rearrange series from the ground model in particular ways, thereby obtaining consistency results going beyond those that follow from comparisons with familiar cardinal characteristics. Finally, we deal briefly with some variants concerning rearrangements by a special sort of permutation and with rearranging some divergent series to become (conditionally) convergent. The ultrafilter number and $$\mathfrak{hm}$$ https://zbmath.org/1516.03016 2023-09-19T14:22:37.575876Z "Guzmán, Osvaldo" https://zbmath.org/authors/?q=ai:guzman-gonzalez.osvaldo For $$x,y\in 2^\omega$$ let $$\triangle(x,y)$$ be the length of the largest initial common segment of $$x$$ and $$y$$. The coloring $$c_{\min}:[2^\omega]^2\rightarrow 2$$ is defined by: $$c_{\min}(x,y)=0$$ if $$\triangle(x,y)$$ is even and $$c_{\min}(x,y)=1$$ otherwise. $$\mathfrak{hm}$$ is the smallest cardinality of a family of $$c_{\min}$$-monochromatic sets that covers $$2^\omega$$. It is known that $$\mathfrak{hm}\geq\mathrm{cof}(\mathcal{N})$$ and $$\mathfrak{hm}^+\geq\mathfrak{c}$$, so $$\mathfrak{hm}$$ is high among the cardinal invariants of the Cichon diagram. For an ultrafilter $$\mathcal U$$ on $$\omega$$, let $$\mathbb{S}(\mathcal{U})$$ be the forcing that ultradestroys $$\mathcal U$$, as introduced in [\textit{S. Shelah}, Arch. Math. Logic 31, No. 6, 433--443 (1992; Zbl 0785.03029)]. Here it is reproved that $$\mathbb{S}(\mathcal{U})$$ is proper (Proposition 27) and has the Sacks property (Proposition 31). Shelah's model is obtained by a countable support iteration of length $$\omega_2$$ of such forcing notions, and in Sections 5 and 6 it is proved that the property of $$c_{\min}$$-covering is preserved in each step of the iteration. The main result is the following. Theorem 84. In the Shelah's model holds $$\mathfrak{hm}=\omega_1$$ and $$\mathfrak{u}=\omega_2$$. In particular, $$\mathfrak{hm}<\mathfrak{u}$$ is relativelly consistent with the axioms of ZFC. Hrušak's $$\diamondsuit_\mathfrak{d}$$ principle is the following axiom: There is a sequence $$\langle d_\alpha:\alpha<\omega_1\rangle$$ where $$d_\alpha:\alpha\rightarrow\omega$$ is such that for every $$f:\omega_1\rightarrow\omega$$ the set $$\{\alpha>\omega:f\upharpoonright\alpha\leq^*d_\alpha\}$$ is nonempty. Theorem 87. $$\diamondsuit_\mathfrak{d}$$ holds in the Shelah's model. Since $$\diamondsuit_\mathfrak{d}$$ implies $$\mathfrak{a}=\omega_1$$, this equality also holds in the Shelah's model. Reviewer: Boris Šobot (Novi Sad) On Russell typicality in set theory https://zbmath.org/1516.03017 2023-09-19T14:22:37.575876Z "Kanovei, Vladimir" https://zbmath.org/authors/?q=ai:kanovei.vladimir-g "Lyubetsky, Vassily" https://zbmath.org/authors/?q=ai:lyubetsky.vassily-a \textit{A. Tzouvaras} [Notre Dame J. Formal Logic 63, No. 2, 185--196 (2022; Zbl 07556130)] defines a set to be \textit{nontypical} (in the sense of Russell) if it belongs to a countable ordinal definable set. Let $$\mathbf{HNT}$$ the class of hereditarily nontypical sets. In this paper, the authors prove the relative consistency of each of the following statements with ZFC: \begin{itemize} \item[1.] $$\mathbf{HOD}=\mathbf{HNT} \varsubsetneq \mathbf{V}$$; \item[2.] $$\mathbf{HOD} \varsubsetneq \mathbf{HNT}=\mathbf{V}$$; \item[3.] $$\mathbf{HOD} \varsubsetneq \mathbf{HNT} \varsubsetneq \mathbf{V}$$; \item[4.] AC does not hold in $$\mathbf{HNT}$$. \end{itemize} For the first item, the authors rely on their previous result [Arch. Math. Logic 57, No. 3--4, 285--298 (2018; Zbl 06860714)] that countable $$\mathbf{OD}$$ sets of reals are preserved in extensions by one Cohen real $$a$$. In this case, the models are $$\mathbf{HOD} = L$$ and $$\mathbf{V}=L[a]$$. For the second item, they use a forcing with perfect trees as in [Arch. Math. Logic 54, No. 5--6, 711--723 (2015; Zbl 1343.03040)] to add to $$L$$ a non-$$\mathbf{OD}$$ real $$a$$ whose Vitali $$E_0$$-equivalence class is (countable) $$\mathbf{OD}$$. The choice of models is symbolically the same as in the previous item. The third item follows using a finite support, countable product of a variant of Jensen forcing appearing in [the authors, Math. Notes 102, No. 3, 338--349 (2017; Zbl 1420.03130); translation from Mat. Zametki 102, No. 3, 369--382 (2017)]. This product adds a countable sequence $$\mathbf{a}$$ of generic non-$$\mathbf{OD}$$ reals to $$L$$, but the set $$W$$ formed by them is indeed $$\mathbf{OD}$$ in $$\mathbf{V}=L[\mathbf{a}]$$. The model $$\mathbf{V}$$ of the last paragraph also solves the last item, by showing that $$W\in \mathbf{H N T}$$ is not well-orderable there. This proof contains the main technical points of the paper, which involves (among other things) a method introduced by the authors in [Arch. Math. Logic 57, No. 3--4, 285--298 (2018; Zbl 06860714)], and observing that $$\mathbf{a}$$ is $$L(W)$$-generic for the poset of $$1$$-$$1$$ members of $$\mathop{\mathrm{Fn}}(\omega,W)$$. Reviewer: Pedro Sánchez Terraf (Córdoba) Some notions on convexity of picture fuzzy sets convexity of picture fuzzy sets https://zbmath.org/1516.03018 2023-09-19T14:22:37.575876Z "Sangodapo, Taiwo" https://zbmath.org/authors/?q=ai:sangodapo.taiwo-olubunmi Summary: The theory of picture fuzzy sets was formulated by \textit{B. C. Cuong} and \textit{V. Kreinovich} [in: Proceedings of the 2013 third world congress on information and communication technologies, WICT 2013, Hanoi, Vietnam, December 15--18, 2013. Piscataway, NJ: IEEE. 1--6 (2013; \url{doi:10.1109/WICT.2013.7113099})] and is considered as one of the best effective tool in decision making problems. In this paper, the concept of picture fuzzy sets was studied. The notion of picture convex fuzzy sets was introduced and some of its properties are presented. Finally, we put forward the picture affine fuzzy sets and investigate some of its characteristics. Some properties of affine intuitionistic fuzzy sets https://zbmath.org/1516.03019 2023-09-19T14:22:37.575876Z "Sangodapo, Taiwo Olubunmi" https://zbmath.org/authors/?q=ai:sangodapo.taiwo-olubunmi "Feng, Yuming" https://zbmath.org/authors/?q=ai:feng.yuming Summary: Intuitionistic fuzzy sets (IFSs) introduced by Atanassov are generalisations of fuzzy sets which are powerful tools in dealing with vagueness. In this paper, concept of convex (concave) IFSs and its characteristics using cut sets of IFSs were studied. In particular, we introduced affine intuitionistic fuzzy sets and investigate some of its characteristics. Embedding $$C^*$$-algebras into the Calkin algebra https://zbmath.org/1516.03020 2023-09-19T14:22:37.575876Z "Farah, Ilijas" https://zbmath.org/authors/?q=ai:farah.ilijas "Katsimpas, Georgios" https://zbmath.org/authors/?q=ai:katsimpas.georgios "Vaccaro, Andrea" https://zbmath.org/authors/?q=ai:vaccaro.andrea Summary: We prove that, under Martin's axiom, every $$\mathrm{C}^\ast$$-algebra of density character less than continuum embeds into the Calkin algebra. Furthermore, we show that it is consistent with Zermelo-Fraenkel set theory plus the axiom of choice, ZFC, that there is a $$\mathrm{C}^\ast$$-algebra of density character less than continuum that does not embed into the Calkin algebra. Pointwise ergodic theorem for locally countable quasi-pmp graphs https://zbmath.org/1516.37008 2023-09-19T14:22:37.575876Z "Tserunyan, Anush" https://zbmath.org/authors/?q=ai:tserunyan.anush Summary: We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an increasing sequence of Borel subgraphs with finite connected components over which the averages of any $$L^1$$ function converges to its expectation. This implies that every (not necessarily pmp) locally countable ergodic Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. A consequence of this is that every ergodic treeable equivalence relation has an ergodic hyperfinite free factor. The pmp case of the main theorem was first proven by R. Tucker-Drob [editorial remark; unpublished] using a deep result from probability theory. Our proof is different: it is self-contained and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant concerning asymptotic averages of functions and a method of tiling a large part of the space with finite sets with prescribed properties. The non-pmp setting additionally exploits a new quasi-order called visibility to analyze the interplay between the Radon-Nikodym cocycle and the graph structure, providing a sufficient condition for hyperfiniteness. Continuity of coordinate functionals of filter bases in Banach spaces https://zbmath.org/1516.46007 2023-09-19T14:22:37.575876Z "de Rancourt, Noé" https://zbmath.org/authors/?q=ai:de-rancourt.noe "Kania, Tomasz" https://zbmath.org/authors/?q=ai:kania.tomasz "Swaczyna, Jarosław" https://zbmath.org/authors/?q=ai:swaczyna.jaroslaw The paper continues the research started in [\textit{T. Kania} and \textit{J. Swaczyna}, Bull. Lond. Math. Soc. 53, No. 1, 231--239 (2021; Zbl 1481.46008)]. Let $$\mathfrak F$$ be a free filter on $$\mathbb N$$. A sequence $$(e_n)_{n \in \mathbb N}$$ in a Banach space $$X$$ is said to be an $$\mathfrak F$$-basis if for every $$x \in X$$ there is a unique sequence of scalars $$(a_n)_{n \in \mathbb N}$$ such that $$\mathfrak F$$-$$\lim_n \sum_{k=1}^n a_k e_k = x$$. It is an open question, asked by the reviewer many years ago, whether the coordinate functionals $$x \mapsto a_n$$ are necessarily continuous. In [loc.\,cit.], the authors gave a positive answer to this question for every projective filter $$\mathfrak{F}$$ under the set-theoretic assumption of the existence of a super-compact cardinal. The paper under review contains an elegant refinement of the previous reasoning that enables the authors, for analytic filters, to avoid the additional set-theoretic assumption, that is, to do everything in the framework of the classical ZFC axiomatics. Let us remark that the class of analytic filters is very wide and includes all filters defined through explicit formulas. In particular, it includes the filter of statistical convergence, which was the original motivation for the study of $$\mathfrak F$$-bases. Apart from this, the authors remark that a basis with respect to an arbitrary filter that has continuous coordinate functionals is also a basis with respect to some other filter that is analytic. So, although the problem remains open in full generality, the class of analytic filters is essential in this setting. Reviewer: Vladimir Kadets (Kharkiv) On a topological Ramsey theorem https://zbmath.org/1516.54002 2023-09-19T14:22:37.575876Z "Kubiś, Wiesław" https://zbmath.org/authors/?q=ai:kubis.wieslaw "Szeptycki, Paul" https://zbmath.org/authors/?q=ai:szeptycki.paul-j Let $$K$$ be a topological space and let $$r$$ be a positive integer. A function $$f:[S]^r\to K$$ is said to be convergent if there is $$p\in K$$ such that for every neighborhood $$U$$ of $$p$$ there is a finite set $$F$$ such that $$f([S\setminus F]^r)\subseteq U$$. A space $$X$$ is said to be an $$r$$-Ramsey space if for every function $$f:[\omega]^r\to X$$ there exists an infinite set $$B\subseteq\omega$$ such that $$f{\restriction}[B]^r$$ is convergent. The authors prove that compact metrizable spaces are $$r$$-Ramsey for all $$r$$. This theorem generalizes the special case for $$r=2$$ proved in [\textit{M. Bojańczyk} et al., Semigroup Forum 85, No. 1, 182--184 (2012; Zbl 1253.05134)]. In the paper under the review, assuming CH, for every $$r\ge1$$ the authors give an example of a compact space that is $$r$$-Ramsey but is not $$(r+1)$$-Ramsey. They also consider the productivity of the $$r$$-Ramsey property and related cardinal invariants. Reviewer: Miroslav Repický (Košice) Locally compact, monotonically normal Dowker space in $$\mathsf{ZF+AD}$$ https://zbmath.org/1516.54007 2023-09-19T14:22:37.575876Z "Burke, Dennis K." https://zbmath.org/authors/?q=ai:burke.dennis-k In a previous paper [\textit{D. K. Burke}, Topology Appl. 312, Article ID 108074, 12 p. (2022; Zbl 1494.54008)] the author constructed a monotonically normal, screenable, hereditarily paracompact space in $$\mathsf{ZFC}$$. This space is a monotonically normal, screenable Dowker space in $$\mathsf{ZF+AD}$$, it is a $$D$$-space in $$\mathsf{ZFC}$$ but not a $$D$$-space in $$\mathsf{ZF+AD}$$. In the paper under review the author constructs a space on the same set $$\omega^\omega\times\omega$$ which is locally compact, monotonically normal, screenable, hereditarily paracompact, and hereditarily a $$D$$-space in $$\mathsf{ZFC}$$. In $$\mathsf{ZF+AD}$$ and in some other models of set theory this space is locally compact, monotonically normal, screenable Dowker and is not a $$D$$-space. Reviewer: Miroslav Repický (Košice) Topological groups with strong disconnectedness properties https://zbmath.org/1516.54015 2023-09-19T14:22:37.575876Z "Sipacheva, Ol'ga" https://zbmath.org/authors/?q=ai:sipacheva.olga-v The author investigates the existence of topological groups whose underlying topological space is basically disconnected, an $$F$$-space, or an $$F'$$-space. After reviewing the history of maximal (disjoint sets have disjoint closures) and extremally disconnected groups she establishes that a finite-dimensional paracompact $$F$$-group of countable pseudocharacter contains an open Boolean subgroup with the same properties. The product $$\mathbb{Z}^{\omega_1}$$ with its $$G_\delta$$-topology is a topological group that is a $$P$$-space and hence basically disconnected. The main result implies a dichotomy for Lindelöf groups that are basically disconnected: either it is a $$P$$-spaces or it has a quotient of countable pseudocharacter and with an open basically disconnected Boolean subgroup. Furthermore, if the free Boolean group over a space $$X$$ is an $$F'$$-space then $$X$$ has at most one non-$$P$$-point; and if there is such an $$X$$ that is not a $$P$$-space then the free Boolean group has an extremally disconnected quotient group. Using older results one obtains that the existence of a free Boolean $$F'$$-group that is not a $$P$$-space is equivalent to the existence of a selective ultrafilter on $$\mathbb{N}$$. The final result shows that the free (abelian) group over a space is an $$F'$$-space iff the space itself is a $$P$$-space. Reviewer: K. P. Hart (Delft)