Mathias forcing and combinatorial covering properties of filters

Abstract

We give topological characterizations of filters $F$ on $w$ such that the Mathias forcing $M_F$ adds no dominating reals or preserves ground model unbounded families. This allows us to answer some questions of Brendle, Guzmán, Hrušák, Mart\'ınez, Minami, and Tsaban.

12 pages, comments are welcome

References in corpus (4)

• Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures
• Scales, fields, and a problem of Hurewicz
• Linear $\sigma$-additivity and some applications
• Hurewicz sets of reals without perfect subsets

Cited by in corpus (13)

• Products of Menger spaces: a combinatorial approach
• Products of Menger spaces in the Miller model
• Mathias--Prikry and Laver type forcing; Summable ideals, coideals, and $+$-selective filters
• Algebra, selections, and additive Ramsey theory
• Products of Hurewicz spaces in the Laver model
• A non-discrete space $X$ with $C_p(X)$ Menger at infinity
• On the structure of Borel ideals in-between the ideals $\mathcal{E}\mathcal{D}$ and $\mathrm{Fin}\otimes \mathrm{Fin}$ in the Katětov order
• Ultrafilters in the random real model
• Covering properties of $\omega$-mad families
• On well-splitting posets
• Menger and consonant sets in the Sacks model
• Coideals as remainders of groups distinguishing between combinatorial covering properties
• Omission of Intervals: Deducing covering properties of subsets of the real line from their combinatorial structure