Ergodic Ramsey theory

Ergodic Ramsey theory is a branch of mathematics where problems motivated by additive combinatorics are proven using ergodic theory.

## Contents

* 1 History
* 2 Szemerédi's theorem
* 3 See also
* 4 References
* 5 Sources

## History

Ergodic Ramsey theory arose shortly after Endre Szemerédi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, when Hillel Furstenberg gave a new proof of this theorem using ergodic theory. It has since produced combinatorial results, some of which have yet to be obtained by other means, and has also given a deeper understanding of the structure of measure-preserving dynamical systems.

## Szemerédi's theorem

Main article: Szemerédi's theorem

Szemerédi's theorem is a result in arithmetic combinatorics, concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured[1] that every set of integers A with positive natural density contains a k term arithmetic progression for every k. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem. Hillel Furstenberg proved the theorem using ergodic principles in 1977.[2]

## See also

* IP set
* Piecewise syndetic set
* Ramsey theory
* Syndetic set
* Thick set

## References

* Ergodic Methods in Additive Combinatorics
* Vitaly Bergelson (1996) Ergodic Ramsey Theory -an update
* Randall McCutcheon (1999). Elemental Methods in Ergodic Ramsey Theory. Springer. ISBN 978-3540668091.

## Sources

1. ↑ Erdős, Paul; Turán, Paul (1936), "On some sequences of integers", Journal of the London Mathematical Society 11 (4): 261–264, doi:10.1112/jlms/s1-11.4.261, http://www.renyi.hu/~p_erdos/1936-05.pdf .
2. ↑ Furstenberg, Hillel (1977), "Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions", Journal d'Analyse Mathématique 31: 204–256, doi:10.1007/BF02813304 .

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