Ideal version of Ramsey’s theorem

Filipów, Rafał; Mrożek, Nikodem; Recław, Ireneusz; Szuca, Piotr

doi:10.1007/s10587-011-0073-3

Cited by 15 publications

(29 citation statements)

References 13 publications

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“…Following is an example of a weakly Ramsey superfilter which is not Ramsey. Essentially the same example was, independently, found by Filipów, Mrożek, Recław, and Szuca [9].…”

Section: Example 12supporting

confidence: 70%

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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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“…Following is an example of a weakly Ramsey superfilter which is not Ramsey. Essentially the same example was, independently, found by Filipów, Mrożek, Recław, and Szuca [9].…”

Section: Example 12supporting

confidence: 70%

“…Recław has informed us of his independent work with Filipów, Mrożek, and Szuca [9], which contains related results, mainly of a descriptive set theoretic flavor.…”

Section: Final Commentsmentioning

confidence: 99%

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

Samet

Tsaban²

2009

Topology and its Applications

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show abstract

“…In particular, it is known that the ideal I d of sets of asymptotic density 0 is not BW, and every F σ ideal is Fin-BW. Moreover, in the paper [8] the authors show how these properties are connected with Ramsey's theorem.…”

Section: Bolzano-weierstrass Propertymentioning

confidence: 91%

Rearrangement of conditionally convergent series on a small set

Filipów

Szuca

2010

Journal of Mathematical Analysis and Applications

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We consider ideals I of subsets of the set of natural numbers such that for every conditionally convergent series n∈ω a n and every r ∈ R there is a permutation π r : ω → ω such that n∈ω a πr (n) = r and n ∈ ω: π r (n) = n ∈ I.We characterize such ideals in terms of extendability to a summable ideal (this answers a question of Wilczyński). Additionally, we consider Sierpiński-like theorems, where one can rearrange only indices with positive a n .

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“…The following notion was suggested by some results in [13,20]. Let us call a scheme a collection {A s : s ∈ 2 <ω } such that A s = A s 0 ∪ A s 1 and A s 0 ∩A s 1 = ∅ for all s ∈ 2 <ω .…”

Section: Terminologymentioning

confidence: 99%