Ideal games, Ramsey sets and the Frechet-Urysohn property

It is shown that Matet's characterization of $\mathcal{H}$-Ramseyness relative to a selective coideal $\mathcal{H}$, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a coideal $\mathcal{H}$ is semiselective if and only if Matet's game-theoretic characterization of $\mathcal{H}$-Ramseyness holds. This gives a game-theoretic counter part to a theorem of Farah, asserting that a coideal $\mathcal{H}$ is semiselective if and only if the family of $\mathcal{H}$-Ramsey subsets of $\mathbb{N}^{[\infty]}$ coincides with the family of those sets having the $Exp(\mathcal{H})$-Baire property. Finally, we show that under suitable assumptions, semiselectivity is equivalent to the Fr\'echet-Urysohn property.