Calculating the closed ordinal Ramsey number Rcl(ω · 2, 3)

Mermelstein, Omer

doi:10.1007/s11856-019-1827-0

Cited by 4 publications

(6 citation statements)

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“…Contrast this upper bound with R cl (ω · 2, 3) ≤ ω 4 · 2 (see Proposition 6.6). Very recently, Omer Mermelstein has produced a draft [Mer17] where, using a careful topological analysis building in part on the ideas from this section, he obtains that R cl (ω · 2, 3) = ω 3 · 2, in particular improving our upper bounds in both the topological and the closed case.…”

Section: The Anti-tree Partial Ordering On Ordinalsmentioning

confidence: 98%

Topological Ramsey numbers and countable ordinals

Caicedo¹,

Hilton²

2017

Foundations of Mathematics

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, on the occasion of his birthday.Abstract. We study the topological version of the partition calculus in the setting of countable ordinals. Let α and β be ordinals and let k be a positive integer. We write β →top (α, k) 2 to mean that, for every red-blue coloring of the collection of 2-sized subsets of β, there is either a red-homogeneous set homeomorphic to α or a blue-homogeneous set of size k. The least such β is the topological Ramsey number R top (α, k).We prove a topological version of the Erdős-Milner theorem, namely that R top (α, k) is countable whenever α is countable. More precisely, we prove that R top (ω ω β , k + 1) ≤ ω ω β·k for all countable ordinals β and finite k. Our proof is modeled on a new easy proof of a weak version of the Erdős-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of α, proving among other results thatOur computations use a variety of techniques, including a topological pigeonhole principle for ordinals, considerations of a tree ordering based on the Cantor normal form of ordinals, and some ultrafilter arguments.

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Section: The Anti-tree Partial Ordering On Ordinalsmentioning

confidence: 98%

Topological Ramsey numbers and countable ordinals

Caicedo¹,

Hilton²

2017

Foundations of Mathematics

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“…In proofs, we shall use this equivalence whenever we need to use that c(i, j; k, ) = c. As was the case with ω-homogeneous colorings, there always exist skeletons for which the induced colorings are canonical. Important remark: In [Mer19], the original definition of a canonical coloring only requires < CB(θ) in Item b and < m k in Item c. However, analyzing the proof of Fact 2, one sees that the proof still goes through for this modified definition (though, one needs to be careful while using this definition since cannot be m k for k = n in the case that γ is not successor and…”

Section: Now Set H(δ) = {δ} ∪ α∈J δ H(α)mentioning

confidence: 99%

“…In this subsection, we shall define the notions of an ω-homogeneous coloring, a normal coloring and a canonical coloring. The latter two notions first appeared in [Mer19].…”

Section: Preliminariesmentioning

confidence: 99%

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On the closed Ramsey numbers Rcl(ω + n, 3)

Kaya

Saglam

2021

Isr. J. Math.

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In this paper, we contribute to the study of topological partition relations for pairs of countable ordinals and prove that, for all integers n ≥ 3,where R cl (•, •) and R(•, •) denote the closed Ramsey numbers and the classical Ramsey numbers, respectively. We also establish the following asymptotically weaker upper bound:eliminating the use of Ramsey numbers. These results improve the previously known upper and lower bounds.

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“…Important remark. In [Mer19], the original definition of a canonical coloring only requires ℓ < CB(θ) in Item b and ℓ < m k in Item c. However, analyzing the proof of Fact 2, one sees that the proof still goes through for this modified definition. (Though, one needs to be careful while using this definition since ℓ cannot be m k for k = n in the case that γ is not successor and…”

Section: Preliminariesmentioning

confidence: 99%

On the closed Ramsey numbers $R^{cl}(ω+n,3)$

Kaya,

Saglam

2020

Preprint

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In this paper, we contribute to the study of topological partition relations for pairs of countable ordinals and prove that, for all integers n ≥ 3,where R cl (•, •) and R(•, •) denote the closed Ramsey numbers and the classical Ramsey numbers respectively. We also establish the following asymptotically weaker upper boundeliminating the use of Ramsey numbers. These results improve the previously known upper and lower bounds.

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Calculating the closed ordinal Ramsey number Rcl(ω · 2, 3)

Abstract: Closed ordinal Ramsey numbers are a topological variant of the classical (ordinal) Ramsey numbers. We compute the exact value of the closed ordinal Ramsey number R cl (ω 2 , 3) = ω 6 .

Cited by 4 publications

References 8 publications

Topological Ramsey numbers and countable ordinals

Topological Ramsey numbers and countable ordinals

On the closed Ramsey numbers Rcl(ω + n, 3)

On the closed Ramsey numbers $R^{cl}(ω+n,3)$

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