Distinguishing groupwise density numbers (Q2463449)

A family \(\mathcal D\) of infinite subsets of \(\omega\) ordered by \(\subseteq^{\ast}\), where \(A\subseteq^{\ast}B\) means \(A\setminus B\) is finite, is said to be groupwise dense if \(\mathcal D\) is downward closed under \(\subseteq^\ast\) and for any partition \(\langle I_n: n\in\omega\rangle\) of \(\omega\) into finite intervals there is a \(B\subseteq\omega\) such that \(\bigcup_{n\in B}I_n\in \mathcal D\). The groupwise density number \(\mathfrak g\) is the minimal size of a family of groupwise dense families with empty intersection. A groupwise dense family \(\mathcal D\) is a groupwise dense ideal if in addition \(\mathcal D\) is closed under finite unions. The groupwise density number \(\mathfrak g_f\) is the minimal size of a family of groupwise dense ideals with empty intersection. Of course \(\mathfrak g\leq\mathfrak g_f\). In this paper the author answers a question of Mildenberger by showing that the inequality \(\mathfrak g <\mathfrak g_f\) is consistent. The author's method employs Laver forcing.