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Weak A2 spaces, the Kastanas game and strategically Ramsey sets

Clement Yung

Abstract

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 $\cal{R}$ 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.

Weak A2 spaces, the Kastanas game and strategically Ramsey sets

Abstract

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 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.
Paper Structure (18 sections, 38 theorems, 32 equations)

This paper contains 18 sections, 38 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Weak A2 spaces
  3. Axioms
  4. Examples
  5. Ramsey sets
  6. The Kastanas game in wA2-spaces
  7. The abstract Kastanas game
  8. Basic Properties
  9. Kastanas Ramsey sets in topological Ramsey spaces
  10. Kastanas Ramsey Sets and the Projective Hierarchy
  11. Projective Hierarchy
  12. Sigma_21̂ Well Ordering
  13. Strategically Ramsey Sets and Gowers Spaces
  14. Gowers Spaces
  15. The Kastanas Game
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose that $(\mathcal{R},r,\leq)$ is a closed triple satisfying A1-A4. Then a set $\mathcal{X} \subseteq \mathcal{R}$ is Kastanas Ramsey iff it is Ramsey.

Theorems & Definitions (105)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2: Natural numbers/Ellentuck space $\lbrack \mathbb{N}\rbrack^\infty$
  • Example 2.3: Infinite block sequences/Gowers' space $\bf{FIN}_k^{[\infty]}$
  • Example 2.4: Infinite block sequences $\bf{FIN}_{\pm k}^{[\infty]}$
  • Example 2.5: Hales-Jewett space $W_{Lv}^{[\infty]}$
  • Example 2.6: Strong subtrees $\mathcal{S}_\infty$
  • Example 2.7: Carlson-Simpson Space $\mathcal{E}^\infty$
  • ...and 95 more