Caicedo, AE and Hilton, JH (2017) Topological Ramsey numbers and countable ordinals. In: Caicedo, AE, Cummings, J, Koellner, PA and Larson, PB, (eds.) Foundations of Mathematics. Contemporary Mathematics, 690 . American Mathematical Society , pp. 87-120. ISBN 978-1-4704-2256-1
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 Rtop(α, k). We prove a topological version of the Erdős-Milner theorem, namely that Rtop(α, k) is countable whenever α is countable. More precisely, we prove that Rtop(ωωβ, k + 1) ≤ ωωβ·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 that Rtop(ω + 1, k + 1) = ωk + 1, Rtop (α, k) < ωω whenever α < ω2, Rtop(ω², k) ≤ ωω and Rtop(ω² + 1, k + 2) ≤ ωω·k + 1 for all finite k.
Metadata
Item Type: | Book Section |
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Authors/Creators: |
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Editors: |
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Keywords: | Partition calculus; countable ordinals |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 23 Sep 2016 15:04 |
Last Modified: | 18 Oct 2017 13:04 |
Published Version: | https://arxiv.org/abs/1510.00078 |
Status: | Published |
Publisher: | American Mathematical Society |
Series Name: | Contemporary Mathematics |
Identification Number: | 10.1090/conm/690 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:104452 |