A note on the van der Waerden complex

TL;DR

This paper analyzes the van der Waerden complex, a mathematical structure based on arithmetic progressions, and classifies its topological properties like vertex decomposability, Cohen-Macaulayness, and shellability using algebraic techniques.

Contribution

It provides a classification of when van der Waerden complexes are vertex decomposable, Cohen-Macaulay, and shellable, advancing understanding of their topological and algebraic properties.

Findings

Classified van der Waerden complexes as vertex decomposable or not Cohen-Macaulay.

Identified conditions under which these complexes are shellable.

Applied combinatorial commutative algebra techniques to topological classification.

Abstract

Ehrenborg, Govindaiah, Park, and Readdy recently introduced the van der Waerden complex, a pure simplicial complex whose facets correspond to arithmetic progressions. Using techniques from combinatorial commutative algebra, we classify when these pure simplicial complexes are vertex decomposable or not Cohen-Macaulay. As a corollary, we classify the van der Waerden complexes that are shellable.