Hypergraph regularity method: Difference between revisions

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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> for alternative versions see [[Terence Tao|Tao]] (2006),<ref>{{Cite journal|last=Tao|first=Terence|authorlink=Terence Tao|date=2006-10-01|title=A variant of the hypergraph removal lemma|journal=[[Journal of Combinatorial Theory]] | series=Series A|language=en|volume=113|issue=7|pages=1257–1280|doi=10.1016/j.jcta.2005.11.006|doi-access=free|issn=0097-3165}}</ref> and [[Timothy Gowers|Gowers]] (2007).<ref name=":1">{{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|doi-access=free}}</ref>

== Definitions ==