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Very informally, the hypergraph regularity lemma decomposes any given <math> k </math>-uniform [[hypergraph]] into a random-like object with bounded parts (with an appropriate boundedness and randomness notions) that is usually easier to work with. On the other hand, the hypergraph counting lemma estimates the number of hypergraphs of a given isomorphism class in some collections of the random-like parts. This is an extension of [[Szemerédi regularity lemma|Szemerédi's regularity lemma]] that partitions any given graph into bounded number parts such that edges between the parts behave almost randomly. Similarly, the hypergraph counting lemma is a generalization of [[Szemerédi regularity lemma#Graph counting lemma|the graph counting lemma]] that estimates number of copies of a fixed graph as a subgraph of a larger graph.
There are several distinct formulations of the method, all of which imply the [[hypergraph removal lemma]] and a number of other powerful results, such as [[Szemerédi's theorem]], as well as some of its multidimensional extensions. The following formulations are due to [[Vojtěch Rödl|V. Rödl]], B. Nagle, J. Skokan, [[Mathias Schacht|M. Schacht]], and [[Yoshiharu Kohayakawa|Y. Kohayakawa]],<ref>{{Cite journal|last=Rödl|first=V.|last2=Nagle|first2=B.|last3=Skokan|first3=J.|last4=Schacht|first4=M.|last5=Kohayakawa|first5=Y.|date=2005-06-07|title=The hypergraph regularity method and its applications|url=https://www.pnas.org/content/102/23/8109|journal=Proceedings of the National Academy of Sciences|language=en|volume=102|issue=23|pages=8109–8113|doi=10.1073/pnas.0502771102|issn=0027-8424|pmc=1149431|pmid=15919821}}</ref>
== Definitions ==
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<blockquote>For all <math> l \geq k \geq 2 </math> and every <math> \mu > 0 </math>, there exists <math> \zeta > 0 </math> and <math> n_0 > 0 </math> so that the following holds. Suppose <math> \mathcal{F}^{(k)} </math> is a <math> k </math>-uniform hypergraph on <math> l </math> vertices and <math> \mathcal{H}^{(k)} </math> is that on <math> n \geq n_0 </math> vertices. If <math> \mathcal{H}^{(k)} </math> contains at most <math> \zeta n </math> copies of <math> \mathcal{F}^{(k)} </math>, then one can delete <math> \mu n^k </math> hyperedges in <math> \mathcal{H}^{(k)} </math> to make it <math> \mathcal{F}^{(k)} </math>-free. </blockquote>One of the original motivations for graph regularity method was to prove [[Szemerédi's theorem]], which states that every dense subset of <math> \mathbb{Z} </math> contains an arithmetic progression of arbitrary length. In fact, by a relatively simple application of [[triangle removal lemma|the triangle removal lemma]], one can prove that every dense subset of <math> \mathbb{Z} </math> contains an arithmetic progression of length 3.
The hypergraph regularity method and hypergraph removal lemma can prove high-dimensional and ring analogues of density version of Szemerédi's theorems, originally proved by Furstenberg and Katznelson.<ref name=":0">{{Cite journal|last=Furstenberg|first=Hillel|last2=Katznelson|first2=Yitzhak|date=1978|title=An ergodic Szemeredi theorem for commuting transformations|journal=J. Analyse Math.|volume=34|pages=275–291}}</ref>
This theorem roughly implies that any dense subset of <math> \mathbb{Z}^d </math> contains any finite pattern of <math> \mathbb{Z}^d </math>. The case when <math> d = 1 </math> and the pattern is arithmetic progression of length some length is equivalent to Szemerédi's theorem.<blockquote>
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