The EGH Conjecture and the Sperner property of complete intersections

TL;DR

This paper explores the connection between the EGH conjecture and the Sperner property in graded complete intersections, proposing that the conjecture's validity would imply the Sperner property for these algebraic structures.

Contribution

It establishes a conditional link showing that if the EGH conjecture holds, then all graded complete intersections possess the Sperner property.

Findings

EGH conjecture implies Sperner property for complete intersections

Conditional proof linking two important algebraic properties

Highlights importance of EGH conjecture in algebraic combinatorics

Abstract

Let $A$ be a graded complete intersection over a field and $B$ the monomial complete intersection with the generators of the same degrees as $A$. The EGH conjecture says that if $I$ is a graded ideal in $A$, then there should be an ideal $J$ in $B$ such that $B/J$ and $A/I$ have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property.