Ideal games and Ramsey sets

Abstract

It is shown that Matet's characterization ([9]) of the Ramsey property relative to a selective co-ideal H, in terms of games of Kastanas ([6]), still holds if we consider semiselectivity ([3]) 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 [3], asserting that a co-ideal H is semiselective if and only if the family of H-Ramsey subsets of N [∞] 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.

Local Ramsey theory: an abstract approach

“…On the other hand, the following result was proved in for the case . Theorem Suppose λ is a weakly compact cardinal.…”

“…This is the case for several interesting examples. With this assumption, the proof of Theorem is almost identical to that of [, Theorem 5.2], so we shall omit the details. Theorem Let be a topological Ramsey space.…”

“…The proof is very similar to that of [10, Theorem 4.8] so we shall leave the details to the reader. On the other hand, the following result was proved in [3] for the case R = N [∞] . Using the results in § 6.2, Theorem 7.6 is generalized to the context of any topological Ramsey space.…”

“…This generalizes the corresponding results for the case when R is equal to Ellentuck's space (cf. [3,10]).…”

“…This generalizes the corresponding results for the case when R is equal to Ellentuck's space (cf. [3,10]). Proposition 4.3 says that the family of H-Ramsey subsets of R together with the family of H-Ramsey null subsets of R is a Marczewski pair.…”