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Mutually enriching connections between ergodic theory and combinatorics - part 7

² The early results of Ramsey theory : Hilbert's irreducibility theorem, Dickson-Schur work on Fermat's equation over finite fields, van der Waerden's theorem, Ramsey's theoremand its rediscovery by Erdos and Szekeres.

² Three main principles of Ramsey theory : First principle: Complete disorder is impossible. Second principle: Behind every 'Partition' result there is a notion of largeness which is responsible for a 'Density' enhancement of this result. Third principle: The sought-after configurations which are always to be found in large sets are abundant.

² Furstenberg's Dynamical approach : Partition Ramsey theory and topological dynamics Dynamical versions of van der Waerden's theorem, Hindman's theorem and Graham-Rothschild-Spencer's geometric Ramsey. Density Ramsey theory and Furstenberg's correspondence principle Furstenberg's correspondence principle. Ergodic Szemeredi's theorem. Polynomial Szemeredi theorem. Density version of the Hales-Jewett theorem.

² Stone-Cech compactifications and Hindman's theorem : Topological algebra in Stone-Cech compactifications. Proof of Hind-man's theorem via Poincare recurrence theorem for ultrafilters.

² IP sets and ergodic Ramsey theory : Applications of IP sets and idempotent ultrafilters to ergodic-theoretical multiple recurrence and to density Ramsey theory. IP-polynomial Szemeredi theorem.

² Open problems and conjectures

If time permits: ² The nilpotent connection, ² Ergodic Ramsey theory and amenable groups

Citation data

• DOI 10.24350/CIRM.V.19072303
• Cite this video Bergelson Vitaly (10/20/16). Mutually enriching connections between ergodic theory and combinatorics - part 7. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19072303
• URL https://dx.doi.org/10.24350/CIRM.V.19072303

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