Van Douwen’s diagram for dense sets of rationals

Abstract

We investigate cardinal invariants related to the structure $\mathrm{Dense}\left(\mathbb{Q}\right)/\mathrm{nwd}$ of dense sets of rationals modulo the nowhere dense sets. We prove that ${\mathfrak{s}}_{\mathbb{Q}}\le min\{\mathfrak{s},\mathrm{add}\left(\mathcal{M}\right)\}$, thus dualizing the already known ${\mathfrak{r}}_{\mathbb{Q}}\ge max\{\mathfrak{r},\mathrm{cof}\left(\mathcal{M}\right)\}$ [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. 183 (2004) 59–80, Theorem 3.6]. We also show the consistency of each of ${\mathfrak{h}}_{\mathbb{Q}}<{\mathfrak{s}}_{\mathbb{Q}}$ and $\mathfrak{h}<{\mathfrak{h}}_{\mathbb{Q}}$. Our results answer four questions of Balcar, Hernández and Hrušák [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. 183 (2004) 59–80, Questions 3.11].