Ideal games and Ramsey sets

It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal H, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a co-ideal H is semiselective if and only if Matet's game-theoretic characterization of the 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 H is semiselective if and only if the family of H-Ramsey subsets of N-[infinity] coincides with the family of those sets having the abstract H-Baire property. Finally, we show that under suitable assumptions, for every semiselective co-ideal H all sets of real numbers are H-Ramsey.