Complexity of Ramsey null sets (Q436231)

Ramsey measurability was introduced by Galvin and Prikry on Borel sets and later generalized by Silver to analytic sets. Ramsey measurable sets are precisely the Baire measurable sets with respect to the Ellentuck topology on \([\omega]^\omega\). In this paper, the author proves two results regarding Ramsey measurable sets.NEWLINENEWLINEThe \(\sigma\)-ideal of countable sets is known to be \({\Pi}^1_1\) on \({\Sigma}^1_1\), i.e., for any Polish space \(Y\) and any \({\Sigma}^1_1\)-set \(A\subseteq Y\times X\) (or a good universal \({\Sigma}^1_1\)-set), \(\{y\in Y: A_y \in {\mathcal A}\}\), the set of codes for \({\Sigma}^1_1\)-sets in \({\mathcal A}\), is in \({\Pi}^1_1\). Moreover, the set of codes for analytic countable sets is \({\Pi}^1_1\)-complete. By a theorem of Hurewicz, for any uncountable Polish space \(X\) the set of countable compact subsets of \(X\) is \({\Pi}^1_1\)-complete in \(K(X)\). As an analogue of Hurewicz's theorem, the author proves that ({Theorem 1}) the set of codes for Ramsey positive \({\Sigma}^1_1\) (or \({G}_\delta\)) sets is \({\Sigma}^1_2\)-complete. Very few natural examples are known for \({\Pi}^1_2\)-completeness, but there are plenty for \({\Pi}^1_1\)-completeness.NEWLINENEWLINEThe proof of Theorem 1 consists of two steps: one is to show that the set of codes for Ramsey positive sets is \(({\Sigma^1_2}, {\Sigma^1_1}\cup{\Pi^1_1})\)-complete, i.e., any \({\Sigma^1_2}\)-set can be reduced to it via a \(({\Sigma^1_1}\cup{\Pi^1_1})\)-submeasurable reduction; and the second step follows from an abstract result ({Theorem 2}): Any \(({\Sigma^1_2}, {\Sigma^1_1}\cup{\Pi^1_1})\)-complete subset of a Polish zero-dimensional space is \({\Sigma^1_2}\)-complete.NEWLINENEWLINEThis paper includes an application of Theorem 1 in the theory of forcing. Zapletal recently developed a general theory of iteration for idealized forcing, and Ikegami developed a general framework for generic absoluteness and transcendence principles for strongly arboreal forcing notions. A key condition in this theory is that the \(\sigma\)-ideal under consideration is ZFC-correct. With respect to the question of Ikegami (and, independently) Pawlikowski and Zapletal, whether these theories can be applied to Mathias forcing (the forcing associated with the \(\sigma\)-ideal of Ramsey null sets), the author gives a negative answer by showing that, as a corollary of Theorem 1 ({Corollary 3}), the \(\sigma\)-ideal of Ramsey null sets is not ZFC-correct.