Weak A2 spaces, the Kastanas game and strategically Ramsey sets

We introduce the notion of a weak A2 space (or wA2-space), which generalises spaces satisfying Todorčević’s axioms A1-A4 and countable vector spaces. We show that in any Polish weak A2 space, analytic sets are Kastanas Ramsey, and discuss the relationship between Kastanas Ramsey sets and sets in the projective hierarchy. We also show that in all spaces satisfying A1-A4, every subset of $\mathcal{R}$ is Kastanas Ramsey iff Ramsey, generalising the recent result by Cano and Di Prisco. Finally, we show that in the setting of Gowers wA2-spaces, Kastanas Ramsey sets and strategically Ramsey sets coincide, providing a connection between the recent studies on topological Ramsey spaces and countable vector spaces.