Hypergraph regularity and the multidimensional Szemerédi theorem


We prove analogues for hypergraphs of Szemerédi’s regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemerédi 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, Rödl, Schacht and Skokan.


W. Timothy Gowers

DPMMS, University of Cambridge, Cambridge CB3 0WB, United Kingdom