Ideal games and Ramsey sets

It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal $\mathcal{H}$, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a co-ideal $\mathcal{H}$ is semiselective if and only if Matet's game-theoretic characterization of the $\mathcal{H}$-Ramsey property holds. This lifts Kastanas's characterization of the classical Ramsey property to its optimal setting, from the point of view of the local Ramsey theory and gives a game-theoretic counterpart to a theorem of Farah \cite{far}, asserting that a co-ideal $\mathcal{H}$ is semiselective if and only if the family of $\mathcal{H}$-Ramsey subsets of $\N^{[\infty]}$ coincides with the family of those sets having the abstract $\mathcal{H}$-Baire property. Finally, we show that under suitable assumptions, for every semiselective co-ideal $\mathcal H$ all sets of real numbers are $\mathcal H$-Ramsey.