Complexity of codes for Ramsey positive sets

TL;DR

This paper extends Sabok's result by establishing conditions under which the set of codes for $G_\delta$ Ramsey positive subsets in topological Ramsey spaces is $\mathbf{\Sigma}^1_2$-complete, highlighting the complexity of such sets.

Contribution

It generalizes Sabok's $\Sigma^1_2$-completeness result to a broader class of topological Ramsey spaces with sufficient conditions.

Findings

Set of codes for $G_\delta$ Ramsey positive subsets can be $\Sigma^1_2$-complete in general topological Ramsey spaces

Provides criteria for $\Sigma^1_2$-completeness in these spaces

Extends understanding of descriptive set-theoretic complexity in Ramsey theory

Abstract

Sabok showed that the set of codes for $G_\delta$ Ramsey positive subsets of $[\omega]^\omega$ is $\mathbf{\Sigma}^1_2$-complete. We extend this result by providing sufficient conditions for the set of codes for $G_\delta$ Ramsey positive subsets of an arbitrary topological Ramsey space to be $\mathbf{\Sigma}^1_2$-complete.