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

Producción científica: Article › revisión exhaustiva

3 Citas (Scopus)

Resumen

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.

Idioma original	English
Páginas (desde-hasta)	287-300
Número de páginas	14
Publicación	Journal of Number Theory
Volumen	172
DOI	https://doi.org/10.1016/j.jnt.2016.08.012
Estado	Published - mar 1 2017

Nota bibliográfica

Publisher Copyright:
© 2016 Elsevier Inc.

Financiación

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.

Financiadores	Número del financiador
Simons Foundation	206001
National Security Agency	H98230-13-1-0280

ASJC Scopus subject areas

Algebra and Number Theory

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title = "The van der Waerden complex",

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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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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SP - 287

EP - 300

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

The van der Waerden complex

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ASJC Scopus subject areas

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