Combinatorics of dense subsets of the rationals
Volume 183 / 2004
Abstract
We study combinatorial properties of the partial order $({\rm Dense}(\Bbb Q ), \subseteq)$. To do that we introduce cardinal invariants $\frak p_\Bbb Q,\frak t_\Bbb Q,\frak h_\Bbb Q,\frak s_\Bbb Q,\frak r_\Bbb Q, \frak i_\Bbb Q$ describing properties of ${\rm Dense}(\Bbb Q )$. These invariants satisfy $\frak p_\Bbb Q\leq \frak t_\Bbb Q\leq \frak h_\Bbb Q\leq \frak s_\Bbb Q \leq\frak r_\Bbb Q\leq\frak i_\Bbb Q$. We compare them with their analogues in the well studied Boolean algebra ${\cal P}(\omega)/{\rm fin}$. We show that $\frak p_\Bbb Q =\frak p$, $\frak t_\Bbb Q =\frak t$ and $\frak i_\Bbb Q =\frak i$, whereas $\frak h_\Bbb Q >\frak h$ and $\frak r_\Bbb Q >\frak r$ are both shown to be relatively consistent with ZFC. We also investigate combinatorics of the ideal nwd of nowhere dense subsets of ,$\Bbb Q$. In particular, we show that $\mathop{\rm non}\nolimits (\mathcal M)=\min \{ \vert \mathcal{D}\vert : \mathcal{D}\subseteq {\rm Dense}( \Bbb{R}) \wedge ( \forall I\in \mathop{\rm nwd}\nolimits (\Bbb R) ) ( \exists D\in \mathcal{D}) ( I\cap D=\emptyset ) \}$ and $\mathop{\rm cof}\nolimits (\mathcal M)=\min \{ \vert \mathcal{D}\vert : \mathcal{D}\subseteq {\rm Dense}( \Bbb{Q}) \wedge ( \forall I\in \mathop{\rm nwd}\nolimits ) ( \exists D\in \mathcal{D}) ( I\cap D=\emptyset ) \}$. We use these facts to show that $\mathop{\rm cof}\nolimits (\mathcal M)\leq\frak i$, which improves a result of S. Shelah.
