Perfect set properties in models of ZF

Carlos Augusto Di Prisco; Franklin C. Galindo

Displaying similar documents to “Perfect set properties in models of ZF”

The $G_{δ}$ -topology and K-analytic spaces without perfect compact sets

Jose L. Blasco (1990)

Colloquium Mathematicae

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Factors of a perfect square

Tsz Ho Chan (2014)

Acta Arithmetica

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We consider a conjecture of Erdős and Rosenfeld and a conjecture of Ruzsa when the number is a perfect square. In particular, we show that every perfect square n can have at most five divisors between $\sqrt n - ∜ n {(l o g n)}^{1 / 7}$ and $\sqrt n + ∜ n {(l o g n)}^{1 / 7}$ .

Perfect set theorems

Otmar Spinas (2008)

Fundamenta Mathematicae

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We study splitting, infinitely often equal (ioe) and refining families from the descriptive point of view, i.e. we try to characterize closed, Borel or analytic such families by proving perfect set theorems. We succeed for $G_{δ}$ hereditary splitting families and for analytic countably ioe families. We construct several examples of small closed ioe and refining families.

Monochromatic kernel-perfectness of special classes of digraphs

Hortensia Galeana-Sánchez, Luis Alberto Jiménez Ramírez (2007)

Discussiones Mathematicae Graph Theory

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In this paper, we introduce the concept of monochromatic kernel-perfect digraph, and we prove the following two results: (1) If D is a digraph without monochromatic directed cycles, then D and each $α_{v}, v \in V (D)$ are monochromatic kernel-perfect digraphs if and only if the composition over D of ${(α_{v})}_{v \in V (D)}$ is a monochromatic kernel-perfect digraph. (2) D is a monochromatic kernel-perfect digraph if and only if for any B ⊆ V(D), the duplication of D over B, $D^{B}$ , is a monochromatic kernel-perfect digraph. ...

On the closure of Baire classes under transfinite convergences

Tamás Mátrai (2004)

Fundamenta Mathematicae

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Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family $f_{α} : X \to Y (α < ω ₁)$ of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set $α < ω ₁ : f_{α} (x) \neq f (x)$ is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of $Σ ⁰_{η}$ sets which can be interesting in its own right.

Functions of Baire class one

Denny H. Leung, Wee-Kee Tang (2003)

Fundamenta Mathematicae

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Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $β (f) \leq ω^{ξ ₁} \cdot ω^{ξ ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...

Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman (2014)

Fundamenta Mathematicae

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Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space $κ^{κ}$ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in $κ^{κ}$ for uncountable regular κ is however consistent (with...

Quasiperfect domination in triangular lattices

Italo J. Dejter (2009)

Discussiones Mathematicae Graph Theory

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A vertex subset S of a graph G is a perfect (resp. quasiperfect) dominating set in G if each vertex v of G∖S is adjacent to only one vertex ( $d_{v}$ ∈ 1,2 vertices) of S. Perfect and quasiperfect dominating sets in the regular tessellation graph of Schläfli symbol 3,6 and in its toroidal quotients are investigated, yielding the classification of their perfect dominating sets and most of their quasiperfect dominating sets S with induced components of the form $K_{ν}$ , where ν ∈ 1,2,3 depends only...

Extension of functions with small oscillation

Denny H. Leung, Wee-Kee Tang (2006)

Fundamenta Mathematicae

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A classical theorem of Kuratowski says that every Baire one function on a $G_{δ}$ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a...

Rudin-like sets and hereditary families of compact sets

Étienne Matheron, Miroslav Zelený (2005)

Fundamenta Mathematicae

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We show that a comeager Π₁¹ hereditary family of compact sets must have a dense $G_{δ}$ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true $F_{σ δ}$ sets.

Finite-to-one continuous s-covering mappings

Alexey Ostrovsky (2007)

Fundamenta Mathematicae

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The following theorem is proved. Let f: X → Y be a finite-to-one map such that the restriction $f | f^{- 1} (S)$ is an inductively perfect map for every countable compact set S ⊂ Y. Then Y is a countable union of closed subsets $Y_{i}$ such that every restriction $f | f^{- 1} (Y_{i})$ is an inductively perfect map.

Possibly there is no uniformly completely Ramsey null set of size $2^{ω}$

Andrzej Nowik (2002)

Colloquium Mathematicae

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We show that under the axiom $C P A_{c u b e}$ there is no uniformly completely Ramsey null set of size $2^{ω}$ . In particular, this holds in the iterated perfect set model. This answers a question of U. Darji.

Extending $n$ times differentiable functions of several variables

Hajrudin Fejzić, Dan Rinne, Clifford E. Weil (1999)

Czechoslovak Mathematical Journal

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It is shown that $n$ times Peano differentiable functions defined on a closed subset of $ℝ^{m}$ and satisfying a certain condition on that set can be extended to $n$ times Peano differentiable functions defined on $ℝ^{m}$ if and only if the $n$ th order Peano derivatives are Baire class one functions.

An independency result in connectification theory

Alessandro Fedeli, Attilio Le Donne (1999)

Commentationes Mathematicae Universitatis Carolinae

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A space is called connectifiable if it can be densely embedded in a connected Hausdorff space. Let $ψ$ be the following statement: “a perfect $T_{3}$ -space $X$ with no more than $2^{𝔠}$ clopen subsets is connectifiable if and only if no proper nonempty clopen subset of $X$ is feebly compact". In this note we show that neither $ψ$ nor $\neg ψ$ is provable in ZFC.

Less than $2^{ω}$ many translates of a compact nullset may cover the real line

Márton Elekes, Juris Steprāns (2004)

Fundamenta Mathematicae

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We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from $c o f () < 2^{ω}$ ) that less than $2^{ω}$ many translates of a compact set of measure zero can cover ℝ.

On Borel reducibility in generalized Baire space

Sy-David Friedman, Tapani Hyttinen, Vadim Kulikov (2015)

Fundamenta Mathematicae

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We study the Borel reducibility of Borel equivalence relations on the generalized Baire space $κ^{κ}$ for an uncountable κ with $κ^{< κ} = κ$ . The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.

On the index of an odd perfect number

Feng-Juan Chen, Yong-Gao Chen (2014)

Colloquium Mathematicae

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Suppose that N is an odd perfect number and $q^{α}$ is a prime power with $q^{α} | | N$ . Define the index $m = σ (N / q^{α}) / q^{α}$ . We prove that m cannot take the form $p^{2 u}$ , where u is a positive integer and 2u+1 is composite. We also prove that, if q is the Euler prime, then m cannot take any of the 30 forms q₁, q₁², q₁³, q₁⁴, q₁⁵, q₁⁶, q₁⁷, q₁⁸, q₁q₂, q₁²q₂, q₁³q₂, q₁⁴ q₂, q₁⁵q₂, q₁²q₂², q₁³q₂², q₁⁴q₂², q₁q₂q₃, q₁²q₂q₃, q₁³q₂q₃, q₁⁴q₂q₃, q₁²q₂²q₃, q₁²q₂²q₃², q₁q₂q₃q₄, q₁²q₂q₃q₄, q₁³q₂q₃q₄, q₁²q₂²q₃q₄, q₁q₂q₃q₄q₅, q₁²q₂q₃q₄q₅,...

A note on perfect matchings in uniform hypergraphs with large minimum collective degree

Vojtěch Rödl, Andrzej Ruciński, Mathias Schacht, Endre Szemerédi (2008)

Commentationes Mathematicae Universitatis Carolinae

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For an integer $k \geq 2$ and a $k$ -uniform hypergraph $H$ , let $δ_{k - 1} (H)$ be the largest integer $d$ such that every $(k - 1)$ -element set of vertices of $H$ belongs to at least $d$ edges of $H$ . Further, let $t (k, n)$ be the smallest integer $t$ such that every $k$ -uniform hypergraph on $n$ vertices and with $δ_{k - 1} (H) \geq t$ contains a perfect matching. The parameter $t (k, n)$ has been completely determined for all $k$ and large $n$ divisible by $k$ by Rödl, Ruci’nski, and Szemerédi in [, submitted]. The values of $t (k, n)$ are very close to $n / 2 - k$ . In fact, the function $t (k, n) = n / 2 - k + c_{n, k}$ ,...