Mathias forcing and combinatorial covering properties of filters (Q2795925)

Let \(\mathscr{F}\) be a filter. The \textit{Mathias forcing \(\mathbb{M}_\mathcal{F}\) associated with \(\mathcal{F}\)} is defined as the set of all pairs \(\langle s, F\rangle\) with \(s\in[\omega]^{{<}\omega}\), \(F\in\mathcal{F}\), and \(\max s<\min F\), ordered by \(\langle s, F\rangle\leq \langle t, G\rangle\) if and only if \(F\subseteq G\), \(s\) is an end-extension of \(t\), and \(s\setminus t\subseteq G\). In this paper, the authors investigate the relationship between the forcing-theoretic property of \(\mathbb{M}_\mathcal{F}\) and the covering properties of \(\mathcal{F}\), such as the Menger, Hurewicz, and Scheepers properties, as a subspace of \(\mathcal{P}(\omega)\).NEWLINENEWLINEFor example, it is proved that \(\mathbb{M}_\mathcal{F}\) does not add dominating reals if and only if \(\mathcal{F}\) is Menger. It is also shown that \(\mathbb{M}_\mathcal{F}\) is almost \(\omega^\omega\)-bounding if and only if \(\mathcal{F}\) is Hurewicz. Here, a forcing notion \(P\) is \textit{almost \(\omega^\omega\)-bounding} if and only if for every \(P\)-name \(\dot{f}\) for a real and \(q\in P\), there exists a \(g:\omega\to\omega\) such that for every \(A\in[\omega]^{\omega}\), there exists a \(q_A\leq q\) such that \(q_A\Vdash g\restriction A\not< \dot{f}\restriction A\). This property is important because \(P\) is almost \(\omega^\omega\)-bounding if and only if it preserves all unbounded families of reals in the ground model as unbounded families in the generic extension. The forward direction is observed by Shelah, and the converse is shown in this paper.