Combinatorics of open covers (VII): Groupability

Kočinac, Ljubiša D.R.; Scheepers, Marion

doi:10.4064/fm179-2-2

Cited by 102 publications

(36 citation statements)

References 18 publications

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“…Proof. Applying [11,Theorem 16] we can find a sequence U n : n ∈ ω of clopen ω-covers of X such that for any sequence V n : n ∈ ω , V n ∈ [U n ] <ω , the union n∈ω V n fails to be a groupable ω-cover of X. Passing to finer ωcovers, if necessary, we may additionally assume that U n+1 is a refinement of U n for all n, and the projection of each element of U 0 onto the 0th coordinate (recall that U 0 is a family of subsets of ω ω ) is finite.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Countable dense homogeneous filters and the Menger covering property

2014

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Section: Proof Of Theoremmentioning

confidence: 99%

Countable dense homogeneous filters and the Menger covering property

2014

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“…If S 1 (Ω(X), Ω(X)) holds, then for each cover U ∈ Ω(X) and each finite coloring of the set [U] 2 , there is in Ω(X) a cover V ⊆ U such that the graph [V] 2 is monochromatic. Scheepers proved a large number of results of this type, including ones jointly with Ljubiša Kočinac and others (e.g., [14,25,20]).…”

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confidence: 81%

Algebra, selections, and additive Ramsey theory

Tsaban

2018

Fund. Math.

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Abstract. Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods include, in addition to Hurewicz's game theoretic characterization of Menger's property, extensions of the classic idempotent theory in the Stone-Čech compactification of semigroups, and of the more recent theory of selection principles. This provides strong versions of the mentioned celebrated theorems, where the monochromatic substructures are large, beyond infinitude, in an analytic sense. Reducing the main theorems to the purely combinatorial setting, we obtain nontrivial consequences concerning uncountable cardinal characteristics of the continuum.The main results, modulo technical refinements, are of the following type (definitions provided in the main text): Let X be a Menger space, and U be an infinite open cover of X. Consider the complete graph, whose vertices are the open sets in X. For each finite coloring of the vertices and edges of this graph, there are disjoint finite subsets F 1 , F 2 , . . . of the cover U whose unions V 1 := F 1 , V 2 := F 2 , . . . have the following properties:(1) The sets n∈F V n and n∈H V n are distinct for all nonempty finite sets F < H.(2) All vertices n∈F V n , for nonempty finite sets F , are of the same color.(3) All edges n∈F V n , n∈H V n , for nonempty finite sets F < H, have the same color. (4) The family {V 1 , V 2 , . . . } is an open cover of X. A self-contained introduction to the necessary parts of the needed theories is provided.

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“…The following result is essentially proved in [23], using an auxiliary result from [13]. In the general form stated here, it is proved in [16]. [23,13,16]).…”

Section: An Application To Topological Selection Principlesmentioning

confidence: 93%

“…In the general form stated here, it is proved in [16]. [23,13,16]). For Ω-Lindelöf spaces, the following are equivalent:…”

Section: An Application To Topological Selection Principlesmentioning

confidence: 93%

Superfilters, Ramsey theory, and van der Waerden's Theorem

Samet

Tsaban²

2009

Topology and its Applications

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Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its variant for ultrafilters on the natural numbers. We use them to confirm a conjecture of Ko\v{c}inac and Di Maio, which is a generalization of a Ramsey theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman-Rado-Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers.Comment: Among other things, the results of this paper imply (using its one-dimensional version) a higher-dimensional version of the Green-Tao Theorem on arithmetic progressions in the primes. The bibliography is now update

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Combinatorics of open covers (VII): Groupability

Cited by 102 publications

References 18 publications

Countable dense homogeneous filters and the Menger covering property

Countable dense homogeneous filters and the Menger covering property

Algebra, selections, and additive Ramsey theory

Superfilters, Ramsey theory, and van der Waerden's Theorem

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