Hypergraph regularity and the multidimensional Szemer\'edi theorem

TL;DR

This paper extends regularity and counting lemmas to hypergraphs, providing a combinatorial proof with explicit bounds for the multidimensional Szemerédi theorem, a significant advance in combinatorics.

Contribution

It introduces hypergraph regularity and counting lemmas and offers the first combinatorial, explicit-bound proof of the multidimensional Szemerédi theorem.

Findings

Hypergraph regularity lemma established

First combinatorial proof with explicit bounds for multidimensional Szemerédi theorem

Independent results by other researchers confirm findings

Abstract

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.