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{{Short description|Mathematical method in extremal graph theory}}
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 regularity lemma|Szemerédi's regularity]] and counting lemmas.
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|
== Definitions ==
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* <math> \mathcal{P}^{(1)} = \{V_i \colon i \in [a_1]\} </math> is equitable vertex partition of <math> V </math>. That is <math> |V_1| \leq \ldots \leq |V_{a_1}| \leq | V_1| + 1 </math> .
* <math> \mathcal{P}^{(j)} </math> partitions <math> \mathcal{K}_j(\mathcal{G}^{(1)}) = K_{a_1}^{(j)}(V_1, \ldots, V_{a_1}) </math> so that if <math> P_1^{(j-1)}, \ldots, P_j^{(j-1)} \in \mathcal{P}^{(j-1)} </math> and <math> \mathcal{K}_j(\cup_{i=1}^jP_i^{(j-1)}) \neq \emptyset </math> then <math> \mathcal{K}_j(\cup_{i=1}^jP_i^{(j-1)}) </math> is partitioned into at most <math> a_j </math> parts, all of which are members <math> \mathcal{P}^{(j)} </math>.
* For all but at most <math> \mu n^k </math> <math> k </math>-tuples <math> K \in \binom{V}{k} </math> there is unique <math> (\delta, \mathbf{d},r) </math>-regular <math> (k, k-1) </math>-complex <math> \mathbf{P} = \{P^{(j)}\}_{j=1}^{k-1} </math> such that <math> P^{(j)} </math> has as members <math> \binom{k}{j} </math> different partition classes from <math> \mathcal{P}^{(j)} </math> and <math> K \in \mathcal{K}_k(P^{(k-1)}) \subset \ldots \subset \mathcal{K}_k(P^{(1)}) </math>.</blockquote>Finally, the following defines what it means for a <math> k </math>-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
== Statements ==
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=== [[Hypergraph removal lemma]] ===
<blockquote>For all <math> l \geq k \geq 2 </math> and every <math> \mu > 0 </math>, there exists <math> \zeta > 0 </math> and <math> n_0 > 0 </math> so that the following holds. Suppose <math> \mathcal{F}^{(k)} </math> is a <math> k </math>-uniform hypergraph on <math> l </math> vertices and <math> \mathcal{H}^{(k)} </math> is that on <math> n \geq n_0 </math> vertices. If <math> \mathcal{H}^{(k)} </math> contains at most <math> \zeta n^l </math> copies of <math> \mathcal{F}^{(k)} </math>, then one can delete <math> \mu n^k </math> hyperedges in <math> \mathcal{H}^{(k)} </math> to make it <math> \mathcal{F}^{(k)} </math>-free. </blockquote>One of the original motivations for graph regularity method was to prove [[Szemerédi's theorem]], which states that every dense subset of <math> \mathbb{Z} </math> contains an arithmetic progression of arbitrary length. In fact, by a relatively simple application of [[triangle removal lemma|the triangle removal lemma]], one can prove that every dense subset of <math> \mathbb{Z} </math> 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.<ref name=":0">{{Cite journal|
This theorem roughly implies that any dense subset of <math> \mathbb{Z}^d </math> contains any finite pattern of <math> \mathbb{Z}^d </math>. The case when <math> d = 1 </math> and the pattern is arithmetic progression of length some length is equivalent to Szemerédi's theorem.<blockquote>
==== Furstenberg and Katznelson Theorem ====
Source:<ref name=":0" /> Let <math> T </math> be a finite subset of <math> \mathbb{R}^d </math> and let <math> \delta > 0 </math> be given. Then there exists a finite subset <math> C \subset \mathbb{R}^d </math> such that every <math> Z \subset C </math> with <math> |Z| > \delta |C| </math> contains a homothetic copy of <math> T </math>. (i.e. set of form <math> z + \lambda T </math>, for some <math> z \in \mathbb{R}^d </math> and <math> t \in \mathbb{R} </math>)
Moreover, if <math> T \subset [-t; t]^d </math> for some <math> t \in \mathbb{N} </math>, then there exists <math> N_0 \in \mathbb{N} </math> such that <math> C = [-N,N]^d </math> has this property for all <math> N \geq N_0 </math>.</blockquote>Another possible generalization that can be proven by the removal lemma is when the dimension is allowed to grow.<blockquote>
==== Tengan, Tokushige, Rödl, and Schacht Theorem ====
Let <math> A </math> be a finite ring. For every <math> \delta > 0 </math>, there exists <math> M_0 </math> such that, for <math> M \geq M_0 </math>, any subset <math> Z \subset A^M </math> with <math> |Z| > \delta |A^M| </math> contains a coset of an isomorphic copy of <math> A </math> (as a left <math> A </math>-module).
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</blockquote>
== References ==
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[[Category:Graph theory]]
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