Van Douwen’s diagram for dense sets of rationals

We investigate cardinal invariants related to the structure Dense ( Q ) / nwd of dense sets of rationals modulo the nowhere dense sets. We prove that s Q ≤ min { s , add ( M ) } , thus dualizing the already known r Q ≥ max { r , cof ( M ) } [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 h Q < s Q and h < h 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].