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'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,[1] for alternative versions see Tao (2006),[2] and Gowers (2007).[3]
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[3] 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 below.
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
.
Hypergraph regularity lemma
[edit]
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
[edit]
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
.
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,
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.[4] 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
[edit]
Source:[4]
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
[edit]
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.