Cohen-Macaulay admissible clutters

TL;DR

This paper investigates the Cohen-Macaulay property of admissible clutters, proving the conjecture for certain cases and providing counterexamples and conditions for higher heights.

Contribution

It proves the Cohen-Macaulay conjecture for uniform admissible clutters of heights 2 and 3, and explores conditions and counterexamples for higher heights.

Findings

Proved the conjecture for heights 2 and 3

Counterexamples for higher heights

Additional conditions for height 4

Abstract

There is a one-to-one correspondence between square-free monomial ideals and clutters, which are also known as simple hypergraphs. It was conjectured that unmixed admissible clutters are Cohen-Macaulay. We prove the conjecture for uniform admissible clutters of heights 2 and 3. For admissible clutters of greater heights, we give a family of examples to show that the conjecture may fail. When the height is 4, we give an additional condition under which unmixed admissible clutters are Cohen-Macaulay.