On Roitman's principles $\mathsf{M}\mathsf{H}$ and $\mathrm{\Delta }$

The Model Hypothesis (abbreviated $\mathsf{M}\mathsf{H}$) and $\mathrm{\Delta }$ are set-theoretic axioms introduced by J. Roitman in her work on the box product problem. Answering some questions of Roitman and Williams on these two principles, we show (1) $\mathsf{M}\mathsf{H}$ implies the existence of $P$-points in ${\omega }^{\ast }$ and is therefore not a theorem of $\mathsf{Z}\mathsf{F}\mathsf{C}$; (2) $\mathsf{M}\mathsf{H}$ also fails in the side-by-side Sacks models; (3) as $\mathrm{\Delta }$ holds in these models, this implies $\mathrm{\Delta }$ is strictly stronger than $\mathsf{M}\mathsf{H}$; (4) furthermore, $\mathrm{\Delta }$ holds in a large class of forcing extensions in which it was not previously known to hold.