Annals of Combinatorics

, Volume 22, Issue 4, pp 711–768 | Cite as

A Multidimensional Szemerédi Theorem in the Primes via Combinatorics

  • Brian Cook
  • Ákos MagyarEmail author
  • Tatchai Titichetrakun


Let A be a subset of positive relative upper density of \(\mathbb {P}^d\), the d-tuples of primes. We present an essentially self-contained, combinatorial argument to show that A contains infinitely many affine copies of any finite set \(F\subseteq \mathbb {Z}^d\). This provides a natural multidimensional extension of the theorem of Green and Tao on the existence of long arithmetic progressions in the primes.


Hypergraph regularity Primes Progressions 

Mathematics Subject Classification

05C65 05C55 11B30 11N13 



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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Brian Cook
    • 2
  • Ákos Magyar
    • 1
    Email author
  • Tatchai Titichetrakun
    • 3
  1. 1.University of GeorgiaAthensUSA
  2. 2.Kent State UniversityKentUSA
  3. 3.University of British ColumbiaVancouverCanada

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