Hypergraph regularity and the multidimensional Szemerédi theorem

Details Hypergraph regularity and the multidimensional Szemerédi theorem Hypergraph regularity and the multidimensional Szemerédi theorem

2007-10-16

arxiv.org/abs/0710.3032v1

Mathematics

Combinatorics

53 pages

Scientific paper

We prove analogues for hypergraphs of Szemer\'edi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemer\'edi 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, R\"odl, Schacht and Skokan.

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