Mariusz Rabus, Saharon Shelah
It is consistent that for every function f:R x R-> R there is an uncountable set A subseteq R and two continuous functions f_0,f_1:D(A)-> R such that f(alpha, beta) in {f_0(alpha, beta),f_1(alpha, beta)} for every (alpha, beta) in A^2, alpha not = beta .
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove that the upper bounds for the consistency strength of certain instances of mutual stationarity considered by Liu-Shelah~\cite{MR1469093} are close to optimal. We also consider some related and, as it turns out, stronger properties.
We prove that there exists a countable family of continuous real functions whose graphs together with their inverses cover an uncountable square, i.e. a set of the form $X\times X$, where $X$ is an uncountable subset of the real line. This extends Sierpiński's theorem from 1919, saying that $S\times S$ can be covered by countably many graphs of functions and inverses of functions if and only if the size of $S$ does not exceed $\aleph_1$. Our result is also motivated by Shelah's study of planar Borel sets without perfect rectangles.
Combinatorial covering properties in an uncountable setting: canonical examples
本文通过推广经典可数构造，在不可数设定下构造了满足Hurewicz、Menger和γ-性质的空间例子，并揭示了多个新开问题。
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…
An optimal extension of the Jensen covering lemma, within the limits imposed by Prikry forcing, is proved. If L[E] is an "iterable" weasel with no measurable cardinals, then either L[E] has "indiscernibles", or every uncountable set of ordinals is contained in a set in L[E] of the same cardinality. (The terms "iterable" and "indiscernibles" are made precise in the paper.) Most importantly, there is no hypothesis explicitly limiting the large cardinals which are consistent in L[E].
The Heine-Borel theorem for uncountable coverings has recently emerged as an interesting and central principle in higher-order Reverse Mathematics and computability theory, formulated as follows: HBU is the Heine-Borel theorem for uncountable coverings given as $\cup_{x\in [0,1]}(x-Ψ(x), x+Ψ(x))$ for arbitrary $Ψ:[0,1]\rightarrow \mathbb{R}^{+}$, i.e. the original formulation going back to Cousin (1895) and Lindelöf (1903). In this paper, we show that HBU is equivalent to its restriction to functions continuous almost everywhere, an elegant robustness result. We also obtain a nice splitting HB…
Perfect set dichotomy theorem in generalized Solovay model
本文在广义Solovay模型中证明了完美集二分定理，并探讨了其在不可数正则基数下的扩展及线性序的三个元素基础定理。
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…
Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem
该研究围绕集合论全域V与HOD的紧性现象，证明了奇异强极限基数的幂集分歧特性，回答了Hayut的问题，并揭示Magidor覆盖定理的最优性。
We improve previous work on the consistency strength of mutually stationary sequences of sets concentrating on points with divergent cofinality building on previous work by Adolf, Cox and Welch. Specifically, we have greatly reduced our reliance on covering properties in the proof. This will allow us to handle sequences in which sets concentrating on points of countable cofinality appear infinitely often. Furthermore we will show that if $κ$ is a Jónsson cardinal with $κ< \aleph_κ$ then the sharp for a model with a strong cardinal exists.
Universal wide Aronszajn tree
本文在弱紧致基数一致性假设下，证明存在一棵包含所有宽 Aronszajn 树同构副本的通用宽 Aronszajn 树。
The Ordering Principle and Higher Dependent Choice
本文为正则不可数基数 $κ$ 下，ZF 与高阶依赖选择及序原理在无选择公理时的一致性提供新证明，并推广至更大模型类。
A consistency theorem for cardinal sequences of length $< ω_3$
本文证明在特定条件下，存在一个散射布尔空间，其基数序列为给定的序列 f。
In 1968, Galvin conjectured that an uncountable poset $P$ is the union of countably many chains if and only if this is true for every subposet $Q \subseteq P$ with size $\aleph_1$. In 1981, Rado formulated a similar conjecture that an uncountable interval graph $G$ is countably chromatic if and only if this is true for every induced subgraph $H \subseteq G$ with size $\aleph_1$. Todorcevic has shown that Rado's Conjecture is consistent relative to the existence of a supercompact cardinal, while the consistency of Galvin's Conjecture remains open. In this paper, we survey and collect a variety …
We investigate the Ramsey theory of continuous pair-colorings on complete, separable metric spaces, and apply the results to the problem of covering a plane by functions. The homogeneity number hm(c) of a pair-coloring c:[X]^2 -> 2 is the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2^omega, c_min and c_max, which satisfy hm(c_min)\le hm(c_max) and prove: 1. For every Polish space X and every continuous pair-coloring c:[X]^2 -> 2 with hm(c) uncountable: hm(c)= hm(c_min) or hm(c)=hm(c_max) 2. There is a model of set theory …
We show that the following two theories are equiconsistent: (T) ZFC, CH and "There is a dense ideal on the first uncountable cardinal such that if j is the generic embedding associated with it then its restriction on ordinals is independent of the generic object is". (S) ZF, ADR and "Theta is a regular cardinal." The main result of this paper is that T implies that the minimal model of S exists. Woodin, in unpublished work, showed that the consistency of S implies the consistency of T. We will also give a proof of this result, which, together with our main theorem, establishes the equiconsiste…
A classical theorem of Hechler asserts that the structure $\left(ω^ω,\le^*\right)$ is universal in the sense that for any $σ$-directed poset P with no maximal element, there is a ccc forcing extension in which $\left(ω^ω,\le^*\right)$ contains a cofinal order-isomorphic copy of P. In this paper, we prove a consistency result concerning the universality of the higher analogue $\left(κ^κ,\le^S\right)$: Theorem. Assume GCH. For every regular uncountable cardinal $κ$, there is a cofinality-preserving GCH-preserving forcing extension in which for every analytic quasi-order Q over $κ^κ$ and every st…
We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of $ω_1$-like models of set theory. Specifically, under the $\diamondsuit$ hypothesis and suitable consistency assumptions, we show that there is a family of $2^{ω_1}$ many $ω_1$-like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive $ω_1$-like model of ZFC that does not embed into its own constructible universe; and there can be an $ω_1$-like model of PA whose struc…
A slalom is a sequence of finite sets of length omega. Slaloms are ordered by coordinatewise inclusion with finitely many exceptions. Improving earlier results of Mildenberger, Shelah and Tsaban, we prove consistency results concerning existence and non-existence of an increasing sequence of a certain type of slaloms which covers a bounded set of functions in the Baire space.
Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set (omega \times omega) is partitioned into two pieces along the diagonal, then one piece meets every horizontal line in a finite set, and the other piece meets each vertical line in a finite set. Such partitions or coverings can exist only when the sets forming the product are of limited size. This paper considers such …
A model for global compactness
本文在超可compact基数一致性的假设下，构造了一个集合论模型，其中每个奇异基数λ上存在一个均匀超滤子，满足对于所有比λ的紧致度大的基数θ，该超滤子是θ-不可分解的，并展示了色数紧致性的实例，推翻Hedetniemi猜想的反例在ZFC中的最优性。
We introduce a new topological generalization of the $σ$-projective hierarchy, not limited to Polish spaces. Earlier attempts have replaced $^ωω$ by $^κκ$, for $κ$ regular uncountable, or replaced countable by $σ$-discrete. Instead we close the usual $σ$-projective sets under continuous images and perfect preimages together with countable unions. The natural set-theoretic axiom to apply is $σ$-projective determinacy, which follows from large cardinals. Our goal is to generalize the known results for $K$-analytic spaces (continuous images of perfect preimages of $^ωω$) to these more general set…
We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals. Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are (definitions provided in the paper): \be \item Every product of a concentrated space with a Hurewicz $\sone(\Ga,\Op)$ space satisfies $\sone(\Ga,\Op)$. On the other hand, assuming \CH{}, for each Sierpi…
All spaces of countable spread can be small
本文证明了在弱可 compact 卡尔文条件下，可数扩张空间的拓扑性质与基数关系
Contents of this issue: Workshops on SPM themes; Second workshop on Coverings, Selections and Games in Topology (SPM05); Analysis and Descriptive Set Theory Workshop; Descriptive set theory: Effective methods, equivalence relations; Analysis: typical functions, level sets structure; Organizational and additional details; Research announcements; Cardinal restrictions on some homogeneous compacta; Filters: Topological congruence relations on groups; Inverse Limits and Function Algebras; Ultrafilters and partial products of infinite cyclic groups; More on regular reduced products; Consistency of …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。