Hypergraph regularity method: Difference between revisions

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Hypergraph regularity method is a powerful tool in [[extremal graph theory]] that refers to the combined application of hypergraph regularity lemma and associated counting lemma. It is a generalization of graph regularity method, which refers to the use of [[Szemerédi regularity lemma|Szemerédi's regularity]] and counting lemmas.

Very informally, hypergraph regularity lemma decomposes any given <math> k </math>-uniform [[hypergraph]] into random-like object with bounded parts (with an appropriate boundedness and randomness notions) that is usually easier to work with. On the other hand, hypergraph counting lemma estimates the number of hypergraphs of 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, andhypergraph counting lemma is a generalization of [[Szemerédi regularity lemma#Graph counting lemma|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 [[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=PMC1149431|pmid=15919821}}</ref>, for alternative versions see [[Terence Tao|Tao]] (2006)<ref>{{Cite journal|last=Tao|first=Terence|date=2006-10-01|title=A variant of the hypergraph removal lemma|url=https://www.sciencedirect.com/science/article/pii/S0097316505002177|journal=Journal of Combinatorial Theory, Series A|language=en|volume=113|issue=7|pages=1257–1280|doi=10.1016/j.jcta.2005.11.006|issn=0097-3165}}</ref>, and [[Timothy Gowers|Gowers]] (2007)<ref>{{Cite journal|last=Gowers|first=William|date=2007-11-01|title=Hypergraph regularity and the multidimensional Szemerédi theorem|url=http://doi.org/10.4007/annals.2007.166.897|journal=Annals of Mathematics|volume=166|issue=3|pages=897–946|doi=10.4007/annals.2007.166.897|issn=0003-486X}}</ref>.