Better closed ultrafilters on \(\mathbb{Q}{}\) (Q1206527)

Using MA for countable posets, the author proves, for each \(n\in\mathbb{N}\), there is a free closed ultrafilter \(\alpha\) on \(\mathbb{Q}\) such that (1) if \(K\) is a closed, scattered subspace of \(\mathbb{Q}\), there is some \(F\in\alpha\) such that \(F\cap K=\emptyset\), and (2) \(\{\beta\): \(\beta\) is an ultrafilter on \(\mathbb{Q}\) and \(\alpha\subseteq\beta\}\) has size \(n\).