Eisenbud–Green–Harris conjecture

Establish that for every homogeneous ideal I in K[x_1,\dots,x_n] containing a regular sequence f_1,\dots,f_\ell of degrees e_1\leq\cdots\leq e_\ell, there exists a lex ideal L such that L+(x_1^{e_1},\dots,x_\ell^{e_\ell}) has the same Hilbert function as I.

Background

The Eisenbud–Green–Harris conjecture generalizes the Clements–Lindström theorem. The latter guarantees the existence of a lex-plus-powers ideal with the same Hilbert function when the contained regular sequence is generated by powers of variables; EGH predicts the analogous result for an arbitrary regular sequence, with the degrees of the powers prescribed by the degrees of that sequence.

The paper proves asymptotic and bounded-degree cases for broad classes of ideals, including ideals of prescribed type and sufficiently large initial degree. It explicitly notes that an all-degree version together with removal of the large-initial-degree hypothesis would settle the conjecture.