Monomial ideals and independence of $\mathrm{I}{\mathrm{\Sigma }}_2$

We show that a miniaturised version of Maclagan’s theorem on monomial ideals is equivalent to $1-\mathrm{C}\mathrm{o}\mathrm{n}(\mathrm{I}{\mathrm{\Sigma }}_2)$ and classify a phase transition threshold for this theorem. This work highlights the combinatorial nature of Maclagan’s theorem.