The van der Waerden complex

Richard Ehrenborg; Likith Govindaiah; Peter S. Park; Margaret Readdy

doi:10.1016/j.jnt.2016.08.012

The van der Waerden complex

Richard Ehrenborg, Likith Govindaiah, Peter S. Park, Margaret Readdy

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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 log⁡k/log⁡log⁡k. Furthermore, we give bounds on n and k which imply that the van der Waerden complex is contractible.

Original language	English
Pages (from-to)	287-300
Number of pages	14
Journal	Journal of Number Theory
Volume	172
DOIs	https://doi.org/10.1016/j.jnt.2016.08.012
State	Published - Mar 1 2017

Bibliographical note

Publisher Copyright:
© 2016 Elsevier Inc.

Keywords

Arithmetic progressions
Discrete Morse theory
Van der Waerden complex

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jnt.2016.08.012

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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 log⁡k/log⁡log⁡k. Furthermore, we give bounds on n and k which imply that the van der Waerden complex is contractible.",

keywords = "Arithmetic progressions, Discrete Morse theory, Van der Waerden complex",

author = "Richard Ehrenborg and Likith Govindaiah and Park, {Peter S.} and Margaret Readdy",

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doi = "10.1016/j.jnt.2016.08.012",

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AU - Ehrenborg, Richard

AU - Govindaiah, Likith

AU - Park, Peter S.

AU - Readdy, Margaret

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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 log⁡k/log⁡log⁡k. Furthermore, we give bounds on n and k which imply that the van der Waerden complex is contractible.

AB - 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 log⁡k/log⁡log⁡k. Furthermore, we give bounds on n and k which imply that the van der Waerden complex is contractible.

KW - Arithmetic progressions

KW - Discrete Morse theory

KW - Van der Waerden complex

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SN - 0022-314X

VL - 172

SP - 287

EP - 300

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

The van der Waerden complex

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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