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1
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Indestructible colourings and rainbow Ramsey theorems

100%
Soukup L.
Fundamenta Mathematicae
|
2009
|
tom 202
|
nr 2
161-180
EN
We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c. It is also consistent that $2^{ω₁}$ is arbitrarily large, and there is a function g establishing $2^{ω₁} ↛ [(ω₁,ω₂)]_{ω₁}$; but there is no uncountable g-rainbow subset of $2^{ω₁}$. We also show that if GCH holds then for each k ∈ ω there is a k-bounded colouring f: [ω₁]² → ω₁ and there are two c.c.c. posets 𝓟 and 𝒬 such that $V^{𝓟}$ ⊨ f c.c.c.-indestructibly establishes $ω₁ ↛ *[(ω₁;ω₁)]_{k-bdd}$, but $V^{𝒬}$ ⊨ ω₁ is the union of countably many f-rainbow sets.
2
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Luzin and anti-Luzin almost disjoint families

64%
Roitman J., Soukup L.
Fundamenta Mathematicae
|
1998
|
tom 158
|
nr 1
51-67
EN
Under $MA_{ω_1}$ every uncountable almost disjoint family is either anti-Luzin or has an uncountable Luzin subfamily. This fails under CH. Related properties are also investigated.
3
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More set-theory around the weak Freese–Nation property

64%
Fuchino S., Soukup L.
Fundamenta Mathematicae
|
1997
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tom 154
|
nr 2
159-176
EN
We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang's Conjecture for $ℵ_ω$, we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding Cohen reals, a lot of ccc complete Boolean algebras of cardinality ≤ λ have the $ℵ_1$-Freese-Nation property provided that $μ^{ℵ_0} = μ$ holds for every regular uncountable μ < λ and the very weak square principle holds for each cardinal $ℵ_0 < μ < λ$ of cofinality ω ((Theorem 15). Finally, we prove that there is no $ℵ_2$-Lusin gap if P(ω) has the $ℵ_1$-Freese-Nation property (Theorem 17)
4
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On splitting infinite-fold covers

51%
Elekes M., Mátrai T., Soukup L.
Fundamenta Mathematicae
|
2011
|
tom 212
|
nr 2
95-127
EN
Let X be a set, κ be a cardinal number and let ℋ be a family of subsets of X which covers each x ∈ X at least κ-fold. What assumptions can ensure that ℋ can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover ℋ: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ℝⁿ by polyhedra and by arbitrary convex sets. We focus on problems with κ infinite. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.
5
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Cardinal sequences of length < ω₂ under GCH

51%
Juhász I., Soukup L., Weiss W.
Fundamenta Mathematicae
|
2006
|
tom 189
|
nr 1
35-52
EN
Let 𝓒(α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put $𝓒_{λ}(α) = {s ∈ 𝓒(α): s(0) = λ = min[s(β): β < α]}$. We show that f ∈ 𝓒(α) iff for some natural number n there are infinite cardinals $λ₀i > λ₁ > ... > λ_{n-1}$ and ordinals $α₀,...,α_{n-1}$ such that $α = α₀ + ⋯ +α_{n-1}$ and $f = f₀⏜f₁⏜...⏜f_{n-1}$ where each $f_i ∈ 𝓒_{λ_i}(α_i)$. Under GCH we prove that if α < ω₂ then (i) $𝓒_{ω}(α) = {s ∈ ^{α}{ω,ω₁}: s(0) = ω}$; (ii) if λ > cf(λ) = ω, $𝓒_{λ}(α) = {s ∈ ^{α}{λ,λ⁺}: s(0) = λ, s^{-1}{λ} is ω₁-closed in α}$; (iii) if cf(λ) = ω₁, $𝓒_{λ}(α) = {s ∈ ^{α}{λ,λ⁺}: s(0) = λ, s^{-1}{λ} is ω-closed and successor-closed in α}$; (iv) if cf(λ) > ω₁, $𝓒_{λ}(α) = ^{α}{λ}$. This yields a complete characterization of the classes 𝓒(α) for all α < ω₂, under GCH.
6
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Cardinal sequences and Cohen real extensions

51%
Juhász I., Shelah S., Soukup L., Szentmiklóssy Z.
Fundamenta Mathematicae
|
2004
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tom 181
|
nr 1
75-88
EN
We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most $(2^{ℵ₀})^V$ levels of size ω. We also give a complete ZFC characterization of the cardinal sequences of regular scattered spaces. Although the classes of regular and of 0-dimensional scattered spaces are different, we prove that they have the same cardinal sequences.
7
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Regular spaces of small extent are ω-resolvable

51%
Juhász I., Soukup L., Szentmiklóssy Z.
Fundamenta Mathematicae
|
2015
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tom 228
|
nr 1
27-46
EN
We improve some results of Pavlov and Filatova, concerning a problem of Malykhin, by showing that every regular space X that satisfies Δ(X) > e(X) is ω-resolvable. Here Δ(X), the dispersion character of X, is the smallest size of a non-empty open set in X, and e(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are ω-resolvable. We also prove that any regular Lindelöf space X with |X| = Δ(X) = ω₁ is even ω₁-resolvable. The question whether regular Lindelöf spaces of uncountable dispersion character are maximally resolvable remains wide open.
8
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First countable spaces without point-countable π-bases

51%
Juhász I., Soukup L., Szentmiklóssy Z.
Fundamenta Mathematicae
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2007
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tom 196
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nr 2
139-149
EN
We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that ∙ there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable π-base (in fact, the minimum order of a π-base of the space can be made arbitrarily large); ∙ if there is a κ-Suslin line then there is a first countable GO-space of cardinality κ⁺ in which the order of any π-base is at least κ; ∙ it is consistent to have a first countable, hereditarily Lindelöf regular space having uncountable π-weight and ω₁ as a caliber (of course, such a space cannot have a point-countable π-base).
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