Complexity of codes for Ramsey positive sets

Abstract

Sabok showed that the set of codes for $G_{\delta }$ Ramsey positive subsets of $[\omega {]}^{\omega }$ is ${\mathbf{\Sigma }}_2^1$-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 ${\mathbf{\Sigma }}_2^1$-complete.