Hypergraph regularity and the multidimensional Szemerédi theorem

  title={Hypergraph regularity and the multidimensional Szemer{\'e}di theorem},
  author={W. T. Gowers},
  journal={Annals of Mathematics},
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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