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

Abstract

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