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

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

Szemerédi's theorem is a result in arithmetic combinatorics, concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured 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.

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" (PDF), Journal of the London Mathematical Society, 11 (4): 261–264, CiteSeerX 10.1.1.101.8225, doi:10.1112/jlms/s1-11.4.261.
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, MR 0498471.

• Ergodic theory
• Ramsey theory