Ideal games and Ramsey sets

TL;DR

This paper extends game-theoretic characterizations of Ramsey properties to semiselective co-ideals, connecting them with Baire properties and showing that under certain conditions, all sets of reals are Ramsey.

Contribution

It generalizes Kastanas's characterization from selectivity to semiselectivity and links it with Baire properties and the universality of Ramsey sets under certain assumptions.

Findings

Semiselectivity characterizes the game-theoretic Ramsey property.

The family of $\\mathcal{H}$-Ramsey sets coincides with those having the $\\mathcal{H}$-Baire property.

Under certain conditions, all sets of reals are $\mathcal{H}$-Ramsey.

Abstract

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.