Hypergraph regularity method

In mathematics, the hypergraph regularity method is a powerful tool in <a href="/facts/Extremal_graph_theory/mdkBX6yN">extremal graph theory</a> that refers to the combined application of the hypergraph regularity lemma and the associated counting lemma. It is a generalization of the graph regularity method, which refers to the use of <a href="/facts/Szemer%C3%A9di_regularity_lemma/dKt8T8y8">Szemerédi's regularity</a> and counting lemmas.
Very informally, the hypergraph regularity lemma decomposes any given 
 
 
 
 k
 
 
 {\displaystyle k}
 
-uniform <a href="/facts/Hypergraph/qIR2o2I3">hypergraph</a> 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 <a href="/facts/Szemer%C3%A9di_regularity_lemma/dKt8T8y8">Szemerédi's regularity lemma</a> 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 <a href="/facts/Szemer%C3%A9di_regularity_lemma/dKt8T8y8">the graph counting lemma</a> 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 <a href="/facts/Hypergraph_removal_lemma/j1fCITWL">hypergraph removal lemma</a> and a number of other powerful results, such as <a href="/facts/Szemer%C3%A9di%27s_theorem/6fjvXcEx">Szemerédi's theorem</a>, as well as some of its multidimensional extensions. The following formulations are due to <a href="/facts/Vojt%C4%9Bch_R%C3%B6dl/8h2dy5dU">V. Rödl</a>, B. Nagle, J. Skokan, <a href="/facts/Mathias_Schacht/1GAb7wQN">M. Schacht</a>, and <a href="/facts/Yoshiharu_Kohayakawa/yWyVKQVx">Y. Kohayakawa</a>, for alternative versions see <a href="/facts/Terence_Tao/Do68JKn6">Tao</a> (2006), and <a href="/facts/Timothy_Gowers/El3xoEmE">Gowers</a> (2007).

Hypergraph regularity method open-in-new

Hypergraph regularity method