Hypergraph regularity method
In mathematics, the hypergraph regularity method is a powerful tool in extremal graph theory 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 Szemerédi's regularity and counting lemmas. Very informally, the hypergraph regularity lemma decomposes any given -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'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 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 V. Rödl, B. Nagle, J. Skokan, M. Schacht, and Y. Kohayakawa, for alternative versions see Tao (2006), and Gowers (2007). Definitions 
In order to state the hypergraph regularity and counting lemmas formally, we need to define several rather technical terms to formalize appropriate notions of pseudo-randomness (random-likeness) and boundedness, as well as to describe the random-like blocks and partitions. Notation

 denotes a -uniform clique on  vertices. is an -partite -graph on vertex partition . is the family of all -element vertex sets that span the clique  in . In particular,  is a complete -partite -graph. The following defines an important notion of relative density, which roughly describes the fraction of -edges spanned by -edges that are in the hypergraph. For example, when , the quantity  is equal to the fraction of triangles formed by 2-edges in the subhypergraph that are 3-edges. Definition [Relative density]. For , fix some classes  of  with . Suppose  is an integer. Let  be a subhypergraph of the induced -partite  graph . Define the relative density   .What follows is the appropriate notion of pseudorandomness that the regularity method will use. Informally, by this concept of regularity, -edges () have some control over -edges (). More precisely, this defines a setting where density of  edges in large subhypergraphs is roughly the same as one would expect based on the relative density alone. Formally,Definition [()-regularity]. Suppose  are positive real numbers and  is an integer. is ()-regular with respect to  if for any choice of classes  and any collection of subhypergraphs  of  satisfying  we have .Roughly speaking, the following describes the pseudorandom blocks into which the hypergraph regularity lemma decomposes any large enough hypergraph. In Szemerédi regularity, 2-edges are regularized versus 1-edges (vertices). In this generalized notion, -edges are regularized versus -edges for all . More precisely, this defines a notion of regular hypergraph called -complex, in which existence of -edge implies existence of all underlying -edges, as well as their relative regularity. For example, if  is a 3-edge then ,, and  are 2-edges in the complex. Moreover, the density of 3-edges over all possible triangles made by 2-edges is roughly the same in every collection of subhypergraphs.Definition [-regular -complex]. An -complex  is a system  of -partite  graphs  satisfying . Given vectors of positive real numbers , , and an integer , we say -complex is -regular if

 For each ,  is -regular with density . For each ,  is ()-regular with respect to .The following describes the equitable partition that the hypergraph regularity lemma will induce. A -equitable family of partition is a sequence of partitions of 1-edges (vertices), 2-edges (pairs), 3-edges (triples), etc. This is an important distinction from the partition obtained by Szemerédi's regularity lemma, where only vertices are being partitioned. In fact, Gowers demonstrated that solely vertex partition can not give a sufficiently strong notion of regularity to imply Hypergraph counting lemma. Definition [-equitable partition]. Let  be a real number,  be an integer, and ,  be vectors of positive reals. Let  be a vector of positive integers and  be an -element vertex set. We say that a family of partitions  on  is -equitable if it satisfies the following:

  is equitable vertex partition of . That is  . partitions  so that if  and   then  is partitioned into at most  parts, all of which are members . For all but at most   -tuples  there is unique -regular -complex  such that  has as members  different partition classes from  and .Finally, the following defines what it means for a -uniform hypergraph to be regular with respect to a partition. In particular, this is the main definition that describes the output of hypergraph regularity lemma bellow.Definition [Regularity with respect to a partition]. We say that a -graph  is -regular with respect to a family of partitions  if all but at most   edges  of  have the property that  and if  is unique -complex for which , then  is  regular with respect to . Statements

Hypergraph regularity lemma 
For all positive real , , and functions ,  for  there exists  and  so that the following holds. For any -uniform hypergraph  on  vertices, there exists a family of partitions  and a vector  so that, for  and  where  for all , the following holds. is a -equitable family of partitions and  for every . is  regular with respect to . Hypergraph counting lemma 
For all integers  the following holds:  and there are integers  and  so that, with , , and ,
 
if  is a -regular  complex with vertex partition  and , then

. Applications 
The main application through which most others follow is the hypergraph removal lemma, which roughly states that given fixed  and large    -uniform hypergraphs, if  contains few copies of , then one can delete few hyperedges in  to eliminate all of the copies of . To state it more formally,

Hypergraph removal lemma 
For all  and every , there exists  and  so that the following holds. Suppose  is a -uniform hypergraph on  vertices and  is that on  vertices. If  contains at most  copies of , then one can delete  hyperedges in  to make it -free. One of the original motivations for graph regularity method was to prove Szemerédi's theorem, which states that every dense subset of  contains an arithmetic progression of arbitrary length. In fact, by a relatively simple application of the triangle removal lemma, one can prove that every dense subset of   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. In fact, this approach yields first quantitative bounds for the theorems. This theorem roughly implies that any dense subset of  contains any finite pattern of . The case when  and the pattern is arithmetic progression of length some length is equivalent to Szemerédi's theorem. Furstenberg and Katznelson Theorem 
Let  be a finite subset of  and let  be given. Then there exists a finite subset  such that every  with  contains a homothetic copy of . (i.e. set of form , for some  and )
 
Moreover, if  for some , then there exists  such that  has this property for all .Another possible generalization that can be proven by the removal lemma is when the dimension is allowed to grow. Tengan, Tokushige, Rödl, and Schacht Theorem 
Let  be a finite ring. For every , there exists  such that, for , any subset  with  contains a coset of an isomorphic copy of  (as a left -module). In other words, there are some  such that , where ,  is an injection. References 

Graph theory