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The space of infinite partitions of $\mathbb{N}$ as a topological Ramsey space
Volume 64, no. 1 (2022), pp. 23–48

Abstract

The Ramsey theory of the space of equivalence relations with infinite quotients defined on the set $\mathbb{N}$ of natural numbers is an interesting field of research. We view this space as a topological Ramsey space $(\mathcal{E}_\infty,\leq, r)$ and present a game theoretic characterization of the Ramsey property of subsets of $\mathcal{E}_{\infty}$. We define a notion of coideal and consider the Ramsey property of subsets of $\mathcal{E}_\infty$ localized on a coideal $\mathcal{H}\subseteq \mathcal{E}_{\infty}$. Conditions a coideal $\mathcal{H}$ should satisfy to make the structure $(\mathcal{E}_{\infty},\mathcal{H},\leq, r)$ a Ramsey space are presented. Forcing notions related to a coideal $\mathcal{H}$ and their main properties are analyzed.