Ideal games and Ramsey sets

Abstract

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