# Hypergraph regularity and the multidimensional Szemerédi theorem

@article{Gowers2007HypergraphRA, title={Hypergraph regularity and the multidimensional Szemer{\'e}di theorem}, author={W. T. Gowers}, journal={Annals of Mathematics}, year={2007}, volume={166}, pages={897-946} }

We prove analogues for hypergraphs of Szemeredi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemeredi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound. Similar results with the same consequences have been obtained independently by Nagle, Rodl, Schacht and Skokan.

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#### References

SHOWING 1-10 OF 14 REFERENCES

Quasirandomness, Counting and Regularity for 3-Uniform Hypergraphs

- Mathematics, Computer Science
- Combinatorics, Probability and Computing
- 2006

A combination of regularity and counting lemmas for 3-uniform hypergraphs gives a new proof of a theorem of Frankl and Rödl, of which Szemerédi's theorem for arithmetic progressions of length 4 is a notable consequence. Expand

The ergodic theoretical proof of Szemerédi's theorem

- Mathematics
- 1982

Partial results were obtained previously by K. F. Roth (1952) who established the existence of arithmetic progressions of length three in subsets of Z of positive upper density, and by E. Szemeredi… Expand

A new proof of Szemerédi's theorem

- Mathematics
- 2001

In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these… Expand

Regularity Lemma for k-uniform hypergraphs

- Mathematics, Computer Science
- Random Struct. Algorithms
- 2004

This paper presents a generalization of Szemeredi's Regularity Lemma to k-uniform hypergraphs and results were recently independently and alternatively obtained by W. T. Gowers. Expand

A variant of the hypergraph removal lemma

- Mathematics, Computer Science
- J. Comb. Theory, Ser. A
- 2006

A self-contained proof of the hypergraph removal lemma is given, which proves a slight strengthening of the result, which will use in a subsequent paper to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes. Expand

Regular Partitions of Graphs

- Mathematics
- 1975

Abstract : A crucial lemma in recent work of the author (showing that k-term arithmetic progression-free sets of integers must have density zero) stated (approximately) that any large bipartite graph… Expand

The counting lemma for regular k-uniform hypergraphs

- Mathematics
- 2006

Szemeredi's Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an e-partite graph with… Expand

Note on a Generalization of Roth’s Theorem

- Mathematics
- 2003

We give a simple proof that for sufficiently large N, every subset of of size[N 2]of size at least δN 2 contains three points of the form (a,b), (a + d, b), (a, b + d).

An ergodic Szemerédi theorem for commuting transformations

- Mathematics
- 1978

The classical Poincar6 recurrence theorem asserts that under the action of a measure preserving transformation T of a finite measure space (X, ~, p.), every set A of positive measure recurs in the… Expand

On a problem of Gowers

- Mathematics
- 2005

We prove that every set of cardinality at least contains a triple of the form , where 0$ SRC=http://ej.iop.org/images/1064-5632/70/2/A07/tex_im_2316_img4.gif/>, 0$… Expand