Weak A2 spaces, the Kastanas game and strategically Ramsey sets

TL;DR

This paper introduces weak A2 spaces, explores their properties related to Ramsey theory, and establishes connections between Kastanas Ramsey sets, projective hierarchy, and topological Ramsey spaces.

Contribution

It generalizes the concept of A2 spaces to weak A2 spaces, and links Kastanas Ramsey sets with strategic Ramsey sets in Gowers wA2-spaces.

Findings

Analytic sets are Kastanas Ramsey in Polish weak A2 spaces.

In spaces satisfying A1-A4, all subsets of R are Kastanas Ramsey iff Ramsey.

Kastanas Ramsey sets coincide with strategically Ramsey sets in Gowers wA2-spaces.

Abstract

We introduce the notion of a weak A2 space (or wA2-space), which generalises spaces satisfying Todor\v{c}evi\'c'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 $\cal{R}$ 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.