Q

TL;DR

This paper constructs Q-sets of various uncountable sizes and demonstrates that their squares may not be Q-sets, addressing a question in set theory about the properties of these special sets.

Contribution

It proves the consistent existence of Q-sets of any uncountable size whose squares are not Q, resolving a question posed by A. Miller.

Findings

Existence of Q-sets of size κ for any uncountable κ

Squares of certain Q-sets are not Q-sets

Addresses a longstanding open question in set theory

Abstract

A Q-set is an uncountable set of reals all of whose subsets are relative $G_\delta$ sets. We prove that, for an arbitrary uncountable cardinal kappa, there is consistently a Q-set of size $\kappa$ whose square is not Q. This answers a question of A. Miller.