Analyzing Weak A2 Spaces, the Kastanas Game, and Strategically Ramsey Sets

Alapfogalmak

The paper introduces the notion of a weak A2 space (wA2-space), which generalizes spaces satisfying Todorčević's axioms A1-A4 and countable vector spaces. It shows that in any Polish wA2-space, analytic sets are Kastanas Ramsey, and discusses the relationship between Kastanas Ramsey sets and the projective hierarchy. It also shows that in all spaces satisfying A1-A4, every subset of R is Kastanas Ramsey if and only if it is Ramsey. Finally, it shows that in the setting of Gowers wA2-spaces, Kastanas Ramsey sets and strategically Ramsey sets coincide.

Kivonat

The paper introduces the concept of a weak A2 space (wA2-space), which generalizes the notion of spaces satisfying Todorčević's axioms A1-A4 and countable vector spaces. The key insights are:

In any Polish wA2-space, every analytic set is Kastanas Ramsey. This is shown by relating the Kastanas game on wA2-spaces to the projective hierarchy.

For spaces satisfying axioms A1-A4, a set is Kastanas Ramsey if and only if it is Ramsey. This generalizes a recent result for selective topological Ramsey spaces.

In Gowers wA2-spaces, the Kastanas Ramsey sets and strategically Ramsey sets coincide, providing a connection between topological Ramsey spaces and countable vector spaces.

The paper presents these three main theorems, along with several supporting lemmas and propositions that establish the set-theoretic properties of Kastanas Ramsey sets in wA2-spaces.

Statisztikák

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Idézetek

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Főbb Kivonatok

Weak A2 spaces, the Kastanas game and strategically Ramsey sets

by Clement Yung : arxiv.org 10-02-2024

https://arxiv.org/pdf/2410.00200.pdf

Mélyebb kérdések

How can the results on Kastanas Ramsey sets be extended to other classes of topological spaces beyond wA2-spaces?

The results concerning Kastanas Ramsey sets can potentially be extended to other classes of topological spaces by exploring the foundational properties that characterize wA2-spaces. One approach is to identify spaces that satisfy similar axioms to those of wA2-spaces, particularly focusing on the axioms A1, A3, and the weakened A2 (wA2). For instance, one could investigate the behavior of Kastanas Ramsey sets in spaces that exhibit a form of selectivity or have a well-defined quasi-ordering structure.
Additionally, the concept of Polish spaces, which are separable completely metrizable topological spaces, could serve as a broader framework. By establishing a connection between the properties of analytic sets in Polish spaces and Kastanas Ramsey sets, one could leverage results from descriptive set theory to extend the findings. This could involve analyzing the projective hierarchy within these spaces and determining how Kastanas Ramsey sets behave under various topological transformations or embeddings.
Moreover, exploring the interplay between Kastanas Ramsey sets and other combinatorial properties in spaces like compact Hausdorff spaces or locally compact spaces could yield new insights. The key would be to maintain the essential characteristics of the games defined in the context of Kastanas Ramsey sets while adapting the underlying topological structure.

What are the implications of the connection between Kastanas Ramsey sets and strategically Ramsey sets in Gowers wA2-spaces for applications in areas like functional analysis or combinatorics?

The connection between Kastanas Ramsey sets and strategically Ramsey sets in Gowers wA2-spaces has significant implications for both functional analysis and combinatorics. In functional analysis, the identification of strategically Ramsey sets as a subclass of Kastanas Ramsey sets can inform the study of linear operators and their properties in countable vector spaces. This relationship may lead to new results regarding the stability of certain functional properties under various topological conditions, particularly in the context of weak convergence and compactness.
In combinatorics, the equivalence of Kastanas Ramsey sets and strategically Ramsey sets can enhance our understanding of partition properties and combinatorial games. This connection allows for the application of game-theoretic techniques to analyze the structure of sets within Gowers wA2-spaces, potentially leading to new combinatorial results regarding colorings and partitions of sets. Furthermore, the strategic aspect of these games can provide insights into optimal strategies for players, which can be applied to problems in extremal combinatorics and Ramsey theory.
Overall, the implications of this connection extend to the development of new methodologies for tackling problems in both fields, fostering a deeper understanding of the interplay between topology, combinatorial structures, and functional properties.

Are there any natural examples of wA2-spaces that are not covered by the known classes of topological Ramsey spaces or countable vector spaces, and how would the analysis of Kastanas Ramsey sets in such spaces differ?

Yes, there are natural examples of wA2-spaces that may not fall under the known classes of topological Ramsey spaces or countable vector spaces. One such example could be certain classes of function spaces, particularly those that arise in the context of continuous functions on compact spaces or spaces of sequences with specific convergence properties. These spaces can be constructed to satisfy the axioms A1, wA2, and A3 while lacking the full structure required for them to be classified as topological Ramsey spaces.
The analysis of Kastanas Ramsey sets in these novel wA2-spaces would differ primarily in the nature of the quasi-ordering and the underlying topology. For instance, the properties of continuity and compactness could play a crucial role in determining the behavior of Kastanas Ramsey sets. The interaction between the topology of the space and the combinatorial properties of the sets would need to be carefully examined, potentially leading to different outcomes compared to traditional wA2-spaces.
Moreover, the specific characteristics of the games defined in these spaces might yield unique strategies and outcomes, reflecting the distinct topological properties. This could result in a richer structure of Kastanas Ramsey sets, necessitating the development of new techniques and tools to analyze their properties and relationships with other combinatorial constructs. Overall, exploring these examples could provide valuable insights into the broader landscape of Ramsey theory and its applications in various mathematical domains.