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, asserting that a co-ideal $\mathcal {H}$ 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 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.