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

2020/05/04 by Jamal K. Kawach, Kawach, Jamal K., Stevo Todorčević +1
Computer Science · Mathematics · #03E05 #03E10 #05D10 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2005.01875

openalex publication_date 2020/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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