A Note on Elkin’s Improvement of Behrend’s Construction

  • Ben GreenEmail author
  • Julia Wolf


We provide a short proof of a recent result of Elkin in which large subsets of \(\{1,\ldots,N\}\) free of three-term progressions are constructed.


Arithmetic progressions Roth’s theorem 



The authors are grateful to Tom Sanders for helpful conversations.


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    M. Elkin, An improved construction of progression-free sets, available at
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    R. Graham, On the growth of a van der Waerden-like function, Integers, 6:#A29, 2006Google Scholar
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    B. Landman, A. Robertson and C. Culver, Some new exact van der Waerden numbers, Integers, 5(2):#A10, 2005MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesCambridgeEngland
  2. 2.Mathematical Sciences Research InstituteBerkeleyUSA

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