Let ${{\mathcal J}\,(\mathbb M^2)}$ denote the σ-ideal associated with two-dimensional Miller forcing. We show that it is relatively consistent with ZFC that the additivity of ${{\mathcal J}\,(\mathbb M^2)}$ is bigger than the covering number of the ideal of the meager subsets of ω ω. We also show that Martin’s Axiom implies that the additivity of ${{\mathcal J}\,(\mathbb M^2)}$ is 2 ω .Finally we prove that there are no analytic infinite maximal antichains in any finite product of ${\mathfrak{P}{(\omega)}/{\rm fin}}$.