This note gives an informal overview of the proof in our paper "Borel Conjecture and Dual Borel Conjecture".

The notion of strong measure zero is studied in the context of Polish groups and general separable metric spaces. An extension of a theorem of Galvin, Mycielski and Solovay is given, whereas the theorem is shown to fail for the Baer-Specker group Z ω . The uniformity number of the ideal of strong measure zero subsets of a separable metric space is examined, providing solutions to several problems of Miller and Steprāns (Ann Pure Appl Logic 140(1-3): [52][53][54][55][56][57][58][59] 2006).

We show that the cofinalities of both the Miller ideal m 0 (the σ-ideal naturally related to Miller forcing M) and the Laver ideal ℓ 0 (related to Laver forcing L) are larger than the size of the continuum c in ZFC.

Using a game characterization of distributivity, we show that base matrices for P(ω)/fin of regular height larger than h necessarily have maximal branches which are not cofinal.

By iterating an increasing amoeba for Sacks forcing (implicitly introduced by Louveau, Shelah, and Veličković), we obtain a model in which h (i.e., the distributivity of P(ω)/fin) is smaller than the additivity of the Marczewski ideal (the ideal associated with Sacks forcing). The forcing is different from the usual amoeba for Sacks forcing: Unlike the latter, it has the pure decision and the Laver property, and therefore does not add Cohen reals. In our model, h < hω holds true, which answers a question by Repický who asked whether hω equals h in ZFC.