The van der Waerden complex

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