- Source: Ergodic Ramsey theory
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
Kata Kunci Pencarian:
- Ergodic Ramsey theory
- Ramsey theory
- IP set
- Syndetic set
- Piecewise syndetic set
- Thick set
- List of mathematical theories
- Theory
- Arithmetic combinatorics
- Glossary of areas of mathematics
