It is consistent for every (1 <= n< omega) that (2^omega = omega_n) and there is a function (F:[omega_n]^{< omega}-> omega) such that every finite set can be written at most (2^n-1) ways as the union of two distinct monocolored sets. If GCH holds, for every such coloring there is a finite set that can be written at least (sum^n_{i=1}{n+i choose n}{n choose i}) ways as the union of two sets with the same color.
We study the list-chromatic number and the coloring number of graphs, especially uncountable graphs. We show that the coloring number of a graph coincides with its list-chromatic number provided that the diamond principle holds. Under the GCH assumption, we prove the singular compactness theorem for the list-chromatic number. We also investigate reflection principles for the list-chromatic number and the coloring number of graphs.
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…
In a paper from 1997, Shelah asked whether $Pr_1(λ^+,λ^+,λ^+,λ)$ holds for every inaccessible cardinal $λ$. Here, we prove that an affirmative answer follows from $\square(λ^+)$. Furthermore, we establish that for every pair $χ<κ$ of regular uncountable cardinals, $\square(κ)$ implies $Pr_1(κ,κ,κ,χ)$.
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.
A Q-set is an uncountable set of reals all of whose subsets are relative $G_δ$ sets. We prove that, for an arbitrary uncountable cardinal kappa, there is consistently a Q-set of size $κ$ whose square is not Q. This answers a question of A. Miller.
University of Toronto（多伦多大学）；Institut de Mathématiques de Jussieu, CNRS（巴黎数学研究所，法国国家科学研究中心）；Matematički Institut, SANU（塞尔维亚科学和艺术院数学研究所）；University of North Texas（北德克萨斯大学）
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…
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 construct a nonseparable Banach space $\mathcal X$ (actually, of density continuum) such that any uncountable subset $\mathcal Y$ of the unit sphere of $\mathcal X$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon$ for some $\varepsilon>0$). This solves in the negative the central problem of the search for a nonseparable version of Kottman's theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets o…
We define Magidor cardinals as Jónsson cardinals upon replacing colorings of finite subsets by colorings of $\aleph_0$-bounded subsets. Unlike Jónsson cardinals which appear at some low level of large cardinals, we prove the consistency of having quite large cardinals along with the fact that no Magidor cardinal exists.
The non-commutative Specker phenomenon in the uncountable case
Higman提出并研究了自由积概念的无限形式。设G_i（i在I中）是群，ast_{i in X} G_i是G_i（i在X中）对X子集I的自由积，p_{XY}：ast_{i in Y} G_{i} -> ast_{i in X} G_{i}是X子集q Y子集I的标准同态.（X子集I表示X是I的有限子集。）那么，无限制自由积是逆极限lim（ast_{i in X} G_i，p_{XY}：X子集Y子集I）。我们记为ast_{i in emptyset} G_i= {e}。定理：设F是自由群。然后，对每个同态h：limast G_i-> F，存在可数完全超滤子u_0，. u_m在I上，使得h = h。p_{U_0 cup ... cup U_m}对于u_0，.中的每个U_0 um_m in um_m.若指标集I的基数小于最小可测基数，则存在I的有限子集X_0和同态上交{h}：ast_{i in X_0} G_i-> F使得h=上交{h} .其中p_{X_0}：lim ast G_i-> ast_{i in X_0} G_i是正则投影。
Ramsey's theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey's theorem for pairs due to Erdos and Rado, which states that every 2-coloring of the pairs of rationals has either an infinite 0-homogeneous set or a 1-homogeneous set of order type eta, where eta is the order type of the rationals. This theorem is a natural candidate to lie strictly between the arithmetic comprehension axiom and Ramsey's theorem for pairs. This Erdos-Rado theorem, like the tree theorem for pairs, belongs to a…
The failure of the uncountable non-commutative Specker Phenomenon
Higman在1952年证明了每个自由群都是非交换细长群，也就是说，如果G是一个自由群，h是可数完全自由积（X_omega Z）到G的同态，则存在omega的一个有限子集F和一个同态h：*_{i in F} Z --> G使得h=h rho_F，其中rho_F是从（X_{i in omega}）Z到 *_{i in F}Z的自然映射。对应于阿贝尔的情况下，这种现象被称为非交换斯派克现象。在本文中，我们表明，Higman的结果失败，如果一个通过从可数到不可数。特别地，我们证明了对于非平凡群G_alpha（alpha在lambda中）和不可数基数lambda，从G_alpha的完全自由积到整数环有2个^{2 ^lambda}同态.
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…
In this article, we consider the notion of almost irredundant sets: A subset $\mathcal{X}$ of a C*-algebra $\mathcal{A}$ is called almost irredundant if and only if for every $a\in \mathcal{X}$, the element $a$ does not belong to the norm-closure of $$\{\sum_{i=1}^n λ_i \prod_{j=1}^{n_i}a_{i,j}: \textrm{ where } a_{i,j} \in \mathcal{X}\setminus\{a\} \textrm{ and} \sum |λ_i|\leq 1\}.$$ Since every almost irrredundant set is in particular a discrete set, it follows that the density of $\mathcal{A}$ is an upper bound for the size of almost irredundant sets. We prove that under the Proper Forcing …
Motivated by a problem in additive Ramsey theory, we extend Todorcevic's partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for every Abelian group $G$ of size $\aleph_2$, there exists a coloring $c:G\rightarrow\mathbb Z$ such that for every uncountable $X\subseteq G$ and every integer $k$, there are three distinct elements $x,y,z$ of $X$ such that $c(x+y+z)=k$.
Full reflection of stationary sets at regular cardinals
一个正则不可数基数Kappa的平稳子集S在正则基数上全反射，如果对每个由正则基数组成的高阶平稳集Tsubseteq Kappa，T中存在一个α，使得S cap α是α的平稳子集。我们证明了全反射公理，即每个平稳集在正则基数上全反射，以及n-Mahlo基数的存在性与Pi^1_n-不可描述基数的存在性是等相容的.我们还对极大Mahlo基数给出了适当的推广。
We study Hilbert generated versions of nonseparable Banach spaces $\mathcal X$ considered by Shelah, Steprāns and Wark where the behavior of the norm on nonseparable subsets is so irregular that it does not allow any linear bounded operator on $\mathcal X$ other than a diagonal operator (or a scalar multiple of the identity) plus a separable range operator. We address the questions if these spaces admit uncountable equilateral sets and if their unit spheres admit uncountable $(1+)$-separated or $(1+\varepsilon)$-separated sets. We resolve some of the above questions for two types of these spac…
Using Shelah's revised GCH theorem we prove that if mu<beth_omega <= lambda are cardinals, then every mu-almost disjoint subfamily B of [lambda]^{beth_omega} is essentially disjoint, i.e. for each b from B there is a subset f(b) of b of size < |b| such that the family {b-f(b) b in B} is disjoint. We also show that if mu<=kappa<=lambda, and kappa is infinite, and (x) every mu-almost disjoint subfamily of [lambda]^kappa is essentially disjoint, then (xx) every mu-almost disjoint family B of subsets of lambda with |b|>=kappa for all b from B has a conflict-free colorings with kappa colors. Puttin…
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}.
We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that for every $X\subseteq\mathbb R$ with $|X|=|\mathbb R|$, and every colour $γ\in\mathbb Q$, there are two distinct elements $x_0,x_1$ of $X$ for which $c(x_0+x_1)=γ$. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah. Theorem 2: For every Abelian group $G$, there exists a colouring $c:G\rightarrow\mathbb Q$ such that…
A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of $\ell_1([0,1])$. A wider class of renormings of $\ell_1([0,1])$ which admit no uncountable equilateral sets is also considered.
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…
