Hypergraph regularity method: Difference between revisions

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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's regularity lemma that decomposes any given graph into pseudorandom blocks, namely <math> \varepsilon </math>-regular pairs, and 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=PMC11494311149431|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>. {{AfC submission|t||ts=20211127234938|u=Lepsvera|ns=118|demo=}}

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