Complexity of Ramsey null sets
type: Item
instance of: scholarly article
title: Complexity of Ramsey null sets
zbMATH Open document ID: 1259.03062
author: Marcin Sabok
published in: Advances in Mathematics
publication date: 2012-07-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.5983
reviewed by: Xianghui Shi
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 03E40
zbMATH DE Number: 6059025
zbMATH Keywords: Ramsey null sets
zbMATH Keywords: \(\Sigma^1_2\)-complete sets
zbMATH Keywords: Ramsey measurable sets
zbMATH Keywords: Polish space
zbMATH Keywords: Mathias forcing
zbMATH Keywords: \(\sigma\)-ideal
zbMATH Keywords: ZFC-correct
MaRDI profile type: Publication
OpenAlex ID: W1986518393
arXiv ID: 1003.5983
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DOI: 10.1016/J.AIM.2012.03.001
review text: 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 fo…
description: scientific article; zbMATH DE number 6059025