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Zbl 1159.11005
Tao, Terence
The ergodic and combinatorial approaches to Szemerédi's theorem. (English)
[A] Granville, Andrew (ed.) et al., Additive combinatorics. Providence, RI: American Mathematical Society (AMS). CRM Proceedings and Lecture Notes 43, 145-193 (2007). ISBN 978-0-8218-4351-2/pbk

This is a thorough survey on the various approaches to Szemerédi's theorem on arithmetic progressions in dense sets of integers. Several different proofs have been given: a combinatorial proof by Szemerédi, an ergodic theoretic proof by Furstenberg, proofs based on hypergraphs by Gowers, and independently Nagle, Rödl and Schacht. \par The paper starts with the classical combinatorics, such as Behrend's example, van der Waerden's theorem, and the Hales-Jewett theorem, it continues to topological dynamical systems and Furstenberg's multiple recurrence theorem. The connection to Szemerédi's theorem is elaborated. Then further ergodic theory is introduced and an outline of Furstenberg's proof is given. The paper moves on to the (hyper)-graph theoretic approach, explaining for example the triangle removal lemma and its generalisations. \par Eventually an interesting survey of Szemerédi's original, quite intricate proof [{\it E. Szemerédi}, Acta Arith. 27, 199--245 (1975; Zbl 0303.10056)]. is given. \par Overall, this a an interesting survey which explains and motivates the underlying ideas very well, rather than overloading the reader with all the details. For more details the author refers to the extensive bibliography. Here, the key ideas, and similarities and differences between the various approaches are highlighted.
[Christian Elsholtz (Surrey)]

MSC 2000:: *11B25 Arithmetic progressions
05C55 Generalized Ramsey theory
05C65 Hypergraphs
05C75 Structural characterization of types of graphs
11N13 Primes in progressions
37A45 Relations of ergodic theory with number theory and harmonic analysis
37B20 Notions of recurrence

Keywords: arithmetic progression; Szemerédi's theorem; additive combinatorics; Furstenberg's proof; ergodic theory

Citations: Zbl 0303.10056

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