Gowers, W. T. Hypergraph regularity and the multidimensional Szemerédi theorem. (English) Zbl 1159.05052 Ann. Math. (2) 166, No. 3, 897-946 (2007). In recent years there is huge activity on how to generalize Szemerédi’s Regularity and Removal lemmas for hypergraphs. There are different approaches, different results, even different definitions. (See for example [V. Rödl and J. Skokan, Random Struct. Algorithms 25, No. 1, 1–42 (2004; Zbl 1046.05042), B. Nagle, V. Rödl, and M. Schacht, ibid. 28, 113–179 (2006; Zbl 1093.05045); T. Tao, J. Comb. Theory, Ser. A 113, 1257–1280 (2006; Zbl 1105.05052 ); G. Elek and B. Szegedy, Limits of hypergraphs, regularity and removal lemmas. A non-standard approach. http://arxiv1.library.cornell.edu/abs/0705.2179v1]). The paper under review develops its own versions of the Regularity and Removal lemmas for hypergraphs, and as application it gives the first combinatorial proofs for the multidimensional Szemerédi theorem (and also providing an explicit bound). Reviewer: Péter L. Erdős (Budapest) Cited in 7 ReviewsCited in 153 Documents MSC: 05D10 Ramsey theory 05C65 Hypergraphs 05C30 Enumeration in graph theory 05C35 Extremal problems in graph theory 05C55 Generalized Ramsey theory 05C80 Random graphs (graph-theoretic aspects) 28D15 General groups of measure-preserving transformations Keywords:Regularity Lemma; Removal Lemma; multidimensional Szemerédi theorem; Furstenberg-Katznelson’s theorem Citations:Zbl 1046.05042; Zbl 1093.05045; Zbl 1105.05052 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Link