We give topological characterizations of filters${\cal F}$onωsuch that the Mathias forcing${M_{\cal F}}$adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of Brendle, Guzmán, Hrušák, Martínez, Minami, and Tsaban.

This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We illustrate this interplay by proving that in the Laver model for the consistency of the Borel’s conjecture, the product of any two metrizable spaces with the Hurewicz property has the Menger property.

We prove that if an ultrafilter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{L}}$$\end{document} is not coherent to a Q-point, then each analytic non-σ-bounded topological group G admits an increasing chain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle G_\alpha:\alpha < \mathfrak b(\mathcal L)\rangle}$$\end{document} of its proper subgroups such that: (i) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bigcup_{\alpha}G_\alpha=G}$$\end{document}; and (ii) For every σ-bounded subgroup H of G there exists α such that \documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H\subset G_\alpha}$$\end{document}. In case of the group Sym(ω) of all permutations of ω with the topology inherited from ωω this improves upon earlier results of S. Thomas. (shrink)

For a subset A of a Polish group G, we study the (almost) packing index pack( A) (respectively, Pack( A)) of A, equal to the supremum of cardinalities |S| of subsets $S\subset G$ such that the family of shifts $\{xA\}_{x\in S}$ is (almost) disjoint (in the sense that $|xA\cap yA|<|G|$ for any distinct points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are large in a geometric sense. We show that $\pack}(A)\in\mathbb{N}\cup\{\aleph_0,\mathfrak{c}\}$ for any σ-compact subset A of a (...) Polish group. In each nondiscrete Polish Abelian group G we construct two closed subsets $A,B\subset G$ with $\mathrm{pack}(A)=\mathrm{pack}(B)=\mathfrak{c}$ and \mathrm{Pack}(A\cup B)=1 and then apply this result to show that G contains a nowhere dense Haar null subset $C\subset G$ with pack(C)=Pack(C)=κ for any given cardinal number $\kappa\in[4,\mathfrak{c}]$. (shrink)

We introduce a class of proper posets which is preserved under countable support iterations, includes $\omega ^\omega$ -bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be considered as a possible path towards solving variations of the famous Roitman problem.