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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
https://doi.org/10.33044/revuma.2869
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.
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Published by the Unión Matemática Argentina |
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