Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals

Jamal K. Kawach; Stevo Todorcevic

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arXiv · 2005.01875

Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals

Jamal K. Kawach·Stevo Todorcevic

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Abstract

We define a collection of topological Ramsey spaces consisting of equivalence relations on $\omega$ with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of $\omega$. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) in order to obtain a dual Ramsey theorem for such ordinals.

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BibTeX RIS

Jamal K. Kawach, Stevo Todorcevic. 2020-05-04. Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals. https://arxiv.org/abs/2005.01875

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