Article doi/vol64/v64n1a03 - Revista de la UMA

The space of infinite partitions of

$\mathbb{N}$ as a topological Ramsey space

Julián C. Cano and Carlos A. Di Prisco

Volume 64, no. 1 (2022), pp. 23–48

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.