Complexity of Ramsey null sets

TL;DR

This paper proves that the set of codes for Ramsey positive analytic sets is highly complex, specifically $oldsymbol{ ext{Σ}}^1_2$-complete, revealing deep connections between different forcing notions and the structure of Ramsey null sets.

Contribution

It establishes the $oldsymbol{ ext{Σ}}^1_2$-completeness of the set of codes for Ramsey positive analytic sets, extending the Hurewicz theorem to a higher projective level.

Findings

The set of codes for Ramsey positive analytic sets is $oldsymbol{ ext{Σ}}^1_2$-complete.

The $oldsymbol{ ext{σ}}$-ideal of Ramsey null sets is not ZFC-correct.

The work draws parallels between Sacks and Mathias forcing.

Abstract

We show that the set of codes for Ramsey positive analytic sets is $\mathbf{\Sigma}^1_2$-complete. This is a one projective-step higher analogue of the Hurewicz theorem saying that the set of codes for uncountable analytic sets is $\mathbf{\Sigma}^1_1$-complete. This shows a close resemblance between the Sacks forcing and the Mathias forcing. In particular, we get that the $\sigma$-ideal of Ramsey null sets is not ZFC-correct. This solves a problem posed by Ikegami, Pawlikowski and Zapletal.