The van der Waerden complex

TL;DR

The paper introduces the van der Waerden complex, a simplicial complex based on arithmetic progressions, and studies its topological properties, including homotopy equivalence, cell dimension bounds, and conditions for contractibility.

Contribution

It defines the van der Waerden complex and analyzes its topological structure, providing bounds for contractibility and homotopy equivalence to a CW-complex.

Findings

Homotopy equivalent to a CW-complex with bounded cell dimension

Bounds on n and k imply contractibility of the complex

Asymptotic dimension of cells is at most log k / log log k

Abstract

We introduce the van der Waerden complex ${\rm vdW}(n,k)$ defined as the simplicial complex whose facets correspond to arithmetic progressions of length $k$ in the vertex set $\{1, 2, \ldots, n\}$. We show the van der Waerden complex ${\rm vdW}(n,k)$ is homotopy equivalent to a $CW$-complex whose cells asymptotically have dimension at most $\log k / \log \log k$. Furthermore, we give bounds on $n$ and $k$ which imply that the van der Waerden complex is contractible.