Monomial ideals and independence of IΣ2 DRAFT

We show that a miniaturised version of Maclagan’s theorem on monomial ideals is equivalent to 1−Con(IΣ2) and classify a phase transition threshold for this theorem. This work highlights the combinatorial nature of Maclagan’s theorem. Monomial ideals play an important role in commutative algebra and algebraic combinatorics. Maclagan’s theorem has several applications in computer algebra (See, for example, [8]), so the logical and combinatorial issues which surround it are of particular interest. We rst determine an upper bound for the miniaturised version of Maclagan’s theorem based on a very short proof of Maclagan’s theorem using Friedman’s adjacent Ramsey theorem. We apply known upper bounds for Friedman’s nite adjacent Ramsey theorem for this part. We provide lower bounds for the miniaturised Maclagan theorem. This proof of independence of IΣ2 is a cleaned up version of the proof in [9]. We nish by determining a sharp phase transition threshold for the miniaturised Maclagan’s theorem. These results complement the study by Aschenbrenner and Pong [3] and the determination of lower bounds in [9]. Furthermore this paper ts into the general ∗Partially supported by the Japan Society for the Promotion of Science (KAKENHI 23340020) and a JSPS postdoctoral fellowship for foreign researchers. This is the pre-peer reviewed version of an article which is in press at Mathematical Logic Quarterly