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