Countable partition ordinals

Schipperus, R.

doi:10.1016/j.apal.2009.12.007

Cited by 6 publications

(4 citation statements)

References 7 publications

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“…As we have seen, ω and ω 2 are partition ordinals. Other than these, every countable partition ordinal has the form ω ω β , and in the other direction, ω ω β is a partition ordinal if β has the form ω γ or ω γ + ω δ [Sch10].…”

Section: Questionsmentioning

confidence: 99%

“…Much work has been done to compute these countable ordinal Ramsey numbers. In particular, as announced without proof by Haddad and Sabbagh [HS69c,HS69a,HS69b], there are algorithms for computing R(α, k) for several classes of ordinals α < ω ω and all finite k; details are given in [Cai15] for the case α < ω 2 and in [Mil71] for the case α = ω m for finite m. See also [Wil77,Chapter 7], [HL10], [Sch10] and [Wei14].…”

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confidence: 99%

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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: Questionsmentioning

confidence: 99%

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confidence: 99%

Topological Ramsey numbers and countable ordinals

Caicedo¹,

Hilton²

2017

Foundations of Mathematics

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“…The state of the art regarding the known positive instances of Erdős's problem is the following theorem of Rene Schipperus from his 1999 Ph.D. thesis [40], published many years later, in 2010, as a journal article [41]: (Schipperus 1999). Suppose that β is a countable ordinal whose Cantor Normal Form has at most two summands.…”

Section: Ordinal Partition Relationsmentioning

confidence: 99%

Formalizing Ordinal Partition Relations Using Isabelle/HOL

Džamonja

Koutsoukou-Argyraki

Paulson

2021

Experimental Mathematics

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This is an overview of a formalization project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erd ős-Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all m ∈ N, ω ω −→ (ω ω , m). This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalization process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic.

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“…The state of the art regarding the known positive instances of Erdős's problem is the following theorem of Rene Schipperrus from his 1999 Ph.D. thesis [39], published many years later, in 2010, as a journal paper [40]: (Schipperus 1999) Suppose that β is a countable ordinal whose Cantor Normal Form has at most two summands. Then…”

Section: Ordinal Partition Relationsmentioning

confidence: 99%

Formalising Ordinal Partition Relations Using Isabelle/HOL

Džamonja,

Koutsoukou-Argyraki,

Paulson

2020

Preprint

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This is an overview of a formalisation project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erdős-Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all m ∈ N, ω ω −→ (ω ω , m). This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalisation process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic.

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Countable partition ordinals

Cited by 6 publications

References 7 publications

Topological Ramsey numbers and countable ordinals

Topological Ramsey numbers and countable ordinals

Formalizing Ordinal Partition Relations Using Isabelle/HOL

Formalising Ordinal Partition Relations Using Isabelle/HOL

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