The van der Waerden complex
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2016
- DOI:
- 10.48550/arXiv.1605.00663
- arXiv:
- arXiv:1605.00663
- Bibcode:
- 2016arXiv160500663E
- Keywords:
-
- Mathematics - Combinatorics;
- Mathematics - Number Theory;
- 05;
- 11
- E-Print:
- J. Number Theory, 172 (2017), 287--300