Face Numbers of Certain Cohen-Macaulay Flag Complexes

TL;DR

The paper demonstrates that Cohen-Macaulay complexes nearly balanced can be matched by balanced complexes with identical face vectors, supporting Kalai's conjecture on Cohen-Macaulay flag complexes.

Contribution

It establishes a link between nearly balanced Cohen-Macaulay complexes and balanced ones with the same face vectors, providing evidence for Kalai's conjecture.

Findings

Balanced Cohen-Macaulay complexes can replicate the face vectors of nearly balanced ones.

Supports Kalai's conjecture on the face vectors of Cohen-Macaulay flag complexes.

Provides partial evidence for the conjecture through face vector equivalence.

Abstract

We show that if a $d$-dimensional Cohen-Macaulay complex is, in a certain sense, sufficiently "close" to being balanced, then there is a $d$-dimensional balanced Cohen-Macaulay complex having the same $f$-vector. This in turn provides some partial evidence for a conjecture of Kalai on the $f$-vectors of Cohen-Macaulay flag complexes.