Abstract
We introduce the van der Waerden complex vdW(n,k) defined as the simplicial complex whose facets correspond to arithmetic progressions of length k in the vertex set {1,2,…,n}. We show the van der Waerden complex vdW(n,k) is homotopy equivalent to a CW-complex whose cells asymptotically have dimension at most logk/loglogk. Furthermore, we give bounds on n and k which imply that the van der Waerden complex is contractible.
Original language | English |
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Pages (from-to) | 287-300 |
Number of pages | 14 |
Journal | Journal of Number Theory |
Volume | 172 |
DOIs | |
State | Published - Mar 1 2017 |
Bibliographical note
Funding Information:The authors thank Nigel Pitt for discussions related to asymptotics in Section 3 . The authors also thank the referee for providing the references [1,8,9] . The first author was partially supported by National Security Agency grant H98230-13-1-0280 . This work was partially supported by a grant from the Simons Foundation (# 206001 to Margaret Readdy). The first and fourth authors thank the Princeton University Mathematics Department where this work was initiated.
Publisher Copyright:
© 2016 Elsevier Inc.
Keywords
- Arithmetic progressions
- Discrete Morse theory
- Van der Waerden complex
ASJC Scopus subject areas
- Algebra and Number Theory