Complexity of Ramsey null sets

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.