Granville, Andrew (ed.) et al., Additive combinatorics. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4351-2/pbk). CRM Proceedings and Lecture Notes 43, 145-193 (2007).
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.
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.
Eventually an interesting survey of Szemerédi’s original, quite intricate proof [E. Szemerédi, Acta Arith. 27, 199–245 (1975; Zbl 0303.10056)]. is given.
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.