The Cohen–Macaulay conjecture for unmixed admissible clutters

Let ${\mathcal C}$ be a clutter whose vertices are partitioned into color classes $X^1,\dots,X^d$, and let $e_1,\dots,e_g$ be a partition of its vertex set such that $|e_i\cap X^j|\leq 1$ for all $i,j$, where $g=\operatorname{ht} I({\mathcal C})$. The clutter is admissible if $e_1,\dots,e_g$ are edges and every edge is admissible in the sense that, when represented using the color classes, it has one vertex from each of its first consecutive color classes and nondecreasing matching indices.

The Cohen–Macaulay conjecture. If ${\mathcal C}$ is an admissible clutter and ${\mathcal C}$ is unmixed, then ${\mathcal C}$ is Cohen–Macaulay.

This conjecture seeks to extend the Cohen–Macaulay results known for bipartite graphs with a perfect matching to higher-dimensional admissible clutters. Its resolution status is not determined by the supplied source context.