On a conjecture by Kalai

TL;DR

This paper proves that monomial ideals generated in degree two satisfy a conjecture related to algebraic and combinatorial properties, providing a partial answer to Kalai's conjecture by relating $h$-vectors of certain simplicial complexes.

Contribution

It establishes that $h$-vectors of flag Cohen-Macaulay simplicial complexes are also $h$-vectors of Cohen-Macaulay balanced complexes, advancing understanding of their combinatorial structure.

Findings

Monomial ideals generated in degree two satisfy Eisenbud-Green-Harris conjecture.

$h$-vectors of flag Cohen-Macaulay complexes are realizable as those of balanced complexes.

Provides partial validation of Kalai's conjecture in the context of Cohen-Macaulay complexes.

Abstract

We show that monomial ideals generated in degree two satisfy a conjecture by Eisenbud, Green and Harris. In particular we give a partial answer to a conjecture of Kalai by proving that $h$-vectors of flag Cohen-Macaulay simplicial complexes are $h$-vectors of Cohen-Macaulay balanced simplicial complexes.