Borel Conjecture, dual Borel Conjecture, and other variants of the Borel Conjecture

This survey article is about the Borel Conjecture and several variants (which are inspired by the Galvin-Mycielski-Solovay characterization of strong measure zero) such as the dual Borel Conjecture and the Marczewski Borel Conjecture. We discuss their status in various models of ZFC, and provide the values of the cardinal characteristics in Cichoń’s diagram. In particular, we prove necessary conditions for the (dual) Borel Conjecture, and give a detailed proof of Laver’s theorem that the Borel Conjecture holds in the Laver model and of Carlson’s theorem that the dual Borel Conjecture holds in the Cohen model. Moreover, we give an informal overview of the construction for the model of ZFC in which both the Borel Conjecture and the dual Borel Conjecture hold. We also discuss the notion of very meager (a weakening of strongly meager), and we demonstrate that the (dual) Borel Conjecture is consistent with a projective well-order of the reals.