A new proof of Szemerédi's theorem

@article{Gowers2001ANP,
  title={A new proof of Szemer{\'e}di's theorem},
  author={William T. Gowers},
  journal={Geometric \& Functional Analysis GAFA},
  year={2001},
  volume={11},
  pages={465-588}
}
  • W. T. Gowers
  • Published 15 August 2001
  • Mathematics
  • Geometric & Functional Analysis GAFA
In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these classes contains arbitrarily long arithmetic progressions. This is one of the fundamental results of Ramsey theory, and it has been strengthened in many different directions. A more precise statement of the theorem is as follows. 
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In this paper we introduce and study a certain type of sub semi-group of $\mathbb{R}/\mathbb{Z}$ which turns out to be closely related to \sz's theorem on arithmetic progressions.
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