Abstract
Szemerédi’s regularity lemma is a deep result in graph theory with
applications in many different areas of mathematics. The lemma says
that any graph can be approximated by the union of a bounded number of random-like bipartite graphs and this can be used to extract
the underlying structure of the graph. Recently it has been shown
that there exists polynomial time algorithms that can make this approximation. This survey gives a proof of the regularity lemma, shows
some applications and discusses some algorithmic aspects.
Contents
1 Introduction
1
2 Notations
2
3 The
3.1
3.2
3.3
3.4
Regularity Lemma
Definitions . . . . . . . . . . . .
Szemerédi’s Regularity Lemma .
Proof of the regularity lemma .
Bounds . . . . . . . . . . . . .
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3
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4
10
4 Historical background
12
4.1 van der Waerden’s theorem . . . . . . . . . . . . . . . . . . . 12
4.2 Szemerédi’s theorem . . . . . . . . . . . . . . . . . . . . . . . 12
4.3 The original lemma . . . . . . . . . . . . . . . . . . . . . . . . 14
5 An extremal problem
15
5.1 The regularity graph . . . . . . . . . . . . . . . . . . . . . . . 15
5.2 Erdös-Stone theorem . . . . . . . . . . . . . . . . . . . . . . . 18
6 3-term arithmetic progressions
6.1 Induced matching problem . . . . . . . . . . . . . . . . . . . .
6.2 The (6, 3) problem . . . . . . . . . . . . . . . . . . . . . . . .
6.3 3-term arithmetic progressions . . . . . . . . . . . . . . . . . .
21
21
22
24
7 Algorithmic aspects of the regularity lemma
7.1 Testing for regularity . . . . . . . . . . . . . . . . . . . . . . .
7.2 Creating the regularity partitions . . . . . . . . . . . . . . . .
7.3 Algorithmic applications . . . . . . . . . . . . . . . . . . . . .
26
26
27
29
8 Extending the regularity lemma
30
8.1 The regularity lemma for sparse graphs . . . . . . . . . . . . . 30
8.2 The hypergraph version of the regularity lemma . . . . . . . . 31
9 Recent results using the regularity lemma
33
9.1 Uniform edge distribution and k-universal graphs . . . . . . . 33
9.2 The blow-up lemma . . . . . . . . . . . . . . . . . . . . . . . . 36
A Graphs and hypergraphs
38
B Extremal graph theory
38
i
C Ramsey Theory
40
D NP-completeness
40
ii
1
Introduction
In 1936 the famous mathematicians Paul Erdös and Paul Turán conjectured
P 1
that if A = {a1 , a2 , . . . } is a subset of the natural numbers and
=∞
ai
then A must contain a subset of the form {a, a + b, a + 2b, . . . , a + (n − 1)b}
for some arbitrary large n. This is called an arithmetic progression of length
n. This conjecture appears to be hard to prove and still remains open. There
are however important and interesting special cases of this conjecture. For instance, does the set of all primes contain arithmetic progressions of arbitrary
length? Recently Ben Green and the 2006 Fields medal winner Terrence Tao
showed that this is indeed the case [38]. Their proof basically relied on a
structure theorem that characterises the dichotomy between structure and
randomness. This theorem is called Szemerédi’s regularity lemma.
Szemerédi’s regularity lemma (or uniformity lemma as it is sometimes
called [5]) is an important result which has transformed much of extremal
graph theory (see Appendix B). It can be used as a proof technique with
applications spread throughout the field of combinatorics and its importance
has been realized more and more during the last years. Some examples of
the variety of uses of the regularity lemma can be found in e.g. extremal
graph theory [24, 22, 5, 20], Ramsey theory [31], computer science [22, 3, 10],
general combinatorics, probability theory [37] and functional analysis [28].
The regularity lemma was discovered around 30 years ago as an auxiliary lemma in the proof of another similar conjecture by Erdös and Turán,
concerning arithmetic progressions in so called dense subset of the integers.
The lemma basically says that any graph can be approximated by the union
of a bounded number of random-like bipartite graphs. Its usefulness lies in
that it can, in a way, extract the underlying structure of a graph.
This master thesis is intended as a short survey about the regularity
lemma and its applications in some areas of combinatorics. In Section 3 the
regularity lemma will be stated with a complete proof. Some bounds on
the lemma will also be discussed. Section 4 gives a historical background of
the regularity lemma in connection with additive number theory and Szemerédi’s and van der Waerden’s theorem are mentioned. Section 5 gives an
application of the regularity lemma in extremal graph theory with the proof
of a theorem by Erdös and Stone. Section 6 gives an application in additive
number theory and Roth’s theorem is proved. Section 7 discusses some of the
algorithmic aspects of the regularity lemma. A co-NP-completeness result
is proved and and a constructive version of the regularity lemma is stated.
Section 9 contains a nice application by Yoshiharu Kohayakawa and Vojtech
Rödl regarding uniform edge distribution and universality. A powerful embedding lemma is also mentioned.
1
This thesis should be accessible to anyone with a year or two of mathematical studies at university level behind them. Although not necessary,
it is also recommended that the reader has taken a C or D level course in
graph theory or in combinatorics. At the end of the paper there are four
appendices explaining some basic concepts of graph theory, extremal graph
theory, complexity theory and Ramsey theory.
2
Notations
If not explicitly stated otherwise all graphs in this paper are simple undirected
graphs. To avoid an annoying special case, none of the graphs considered are
the null graph (i.e. the graph whose vertex set is null).
The number of vertices (or the order) of a graph G is denoted by n(G) =
|V (G)| and the number of edges is e(G) = |E(G)|. For A, B ⊂ V (G) we
denote E(A, B) = EG (A, B) as the set of edges that have one endpoint in
A and the other in B. Then we write e(A, B) = eG (A, B) = |E(A, B)|. We
denote the degree of a vertex v by deg(v) and deg(v, A) is the number of
edges from v to vertices in A. Furthermore we write δ(G) = min{deg(v) :
v ∈ V (G)} and ∆(G) = max{deg(v) : v ∈ V (G)}. χ(G) is the chromatic
number of G.
The complete graph on n vertices is denoted by Kn . For the complete
bipartite graph with partitions of size s and t we write Ks,t and the complete
r-partite graph where each partition has s vertices is Ksr .
We denote f ◦n (x) = f ◦ · · · ◦ f (x), i.e. the n times composed function.
| {z }
n
[n] = {1, . . . , n} and for a < b we let [a, b] = {a, a + 1, . . . , b}.
2
3
3.1
The Regularity Lemma
Definitions
Let G be a graph and X, Y ⊂ V (G) be disjoint. We begin by defining the
intuitive concept of density, which is just the actual number of edges between
X and Y , divided by the number of possible edges between them.
Definition 3.1 (Density). The density of the pair (X, Y ) is defined as
d(X, Y ) =
e(X, Y )
|X||Y |
The next definition tells us how uniformly the edges in E(X, Y ) are distributed.
Definition 3.2 (ε-regularity). Let 0 < ε ≤ 1 then the pair (X, Y ) is said
to be ε-regular if for all A ⊂ X and B ⊂ Y satisfying
|A| > ε|X| and |B| > ε|Y |
we have
|d(A, B) − d(X, Y )| < ε
A pair that is not ε-regular is called a witness of the ε-irregularity or
simply ε-irregular. The previous definition gave us the tool to descripe uniformity for a pair of vertex sets. Now we extend this to an entire partition
of the vertex set of a graph.
Definition 3.3 (ε-regular partition). Let P = {Ci }ki=0 be a partition of
V (G) and 0 < ε ≤ 1. C0 is called the exceptional set. Then P is an ε-regular
partition if the following conditions holds:
(i) |C0 | ≤ ε|V (G)|
(ii) |C1 | = |C2 | = · · · = |Ck |
(iii) At most εk 2 of the pairs (Ci , Cj ), 1 ≤ i < j ≤ k are not ε-regular.
The exceptional set C0 can be seen as a storage-place to collect some of
the vertices so that all the other partition sets can have the same size. This
set is disregarded when we check the regularty-condition for the partition
sets. This set is really not necessary at all if we loosen the condition for εregular partitions to partition sets that differ in size by at most one element.
However, for technical reasons it is easier to keep. If a partition satisfies the
second condition, i.e. |C1 | = |C2 | = · · · = |Ck | = l and C0 is the exceptional
set, then we say that the partition is (k, l)-equitable or simply k-equitable if
l is not interesting.
3
3.2
Szemerédi’s Regularity Lemma
Now we state the main theorem1 in this paper.
Theorem 3.4 (Szemerédi’s regularity lemma). For ε > 0 and m ∈ N
there exists an M(ε, m) ∈ N such that for every graph G where n = n(G) ≥
m, there is an ε-regular partition P = {Ci }ki=0 with m ≤ k ≤ M(ε, m).
This basically says that all graphs can be partitioned into a bounded
number of ε-regular partitions, where the upper bound does not depend on
the number of vertices (this is very important). The lower bound can be
used to control certain properties of the regularity partition. More on this in
Section 3.4.
At first glance, this lemma may look innocent, but as we later shall see,
it is a powerful tool with numerous applications.
There are many other different recent versions of this lemma. Some of
them do not even involve graphs and are instead applied on e.g. abelian
groups[18] and in probability theory[37].
3.3
Proof of the regularity lemma
This proof follows the one in [8] closely. First we note the following simple
fact:
Lemma 3.5. Let µ1 , . . . , µk > 0 and γ1 , . . . , γk ≥ 0 then:
P
( γi)2 X γi2
P
≤
µi
µi
(1)
Proof.
This P
is a simple
inequality,
P
P 2 consequence√of the Cauchy-Schwarz
√
( ai bi )2 ≤
a2i
bi , where ai = µi and bi = ei / µi .
Now we define a measure on the uniformity of the pairs in a partition.
Definition 3.6 (Measure of uniformity). Let G be a graph and n = n(G).
Then for disjoint sets A, B ⊂ V (G) we let
q(A, B) =
|A||B| 2
d (A, B)
n2
and if P = {Ci }ki=1 is a partition of V (G) we define
X
q(P) =
q(Ci , Cj )
i<j
1
Due to its importance it will be denoted as a theorem in this paper, even though the
name regularity lemma will be kept for historical reasons.
4
and finally if we have two partitions A and B of A and B respectivly, then
we let
X
q(A, B) =
q(A0 , B 0 )
A0 ∈A
B 0 ∈B
In order to avoid the difficulties that can arise when P = {C0 , C1, . . . , Ck }
where C0 is the exceptional set, we consider C0 as a set of singletons. In other
words we let P̃ = {C1 , . . . , Ck } ∪ {{v} : v ∈ C0 } and define q(P) = q(P̃).
The basic idea for the proof is rather simple. If the partition P of V (G) is
not ε-regular, we just refine this partition to get a new partition P 0 . This will
give a q(P 0 ) that is significantly greater than q(P) and it is easy to show that
the value of q(P) is bounded. So after a bounded numbers of refinements we
will get a ε-regular partition.
Now, let us formalise this. The following two lemmas states that the
value of q will not decrease when we refine a partition.
Lemma 3.7. Let A, B ⊂ V (G) be disjoint. Let A be a partition of A and B
be partition of B, then
q(A, B) ≥ q(A, B)
Proof. Let A = {Ai }ki=1 and B = {Bi }li=1 , then we have
X
q(A, B) =
q(Ai , Bj )
i,j
1 X e2 (Ai , Bj )
=
n2 i,j |Ai ||Bj |
P
2
≥
1 ( i,j e(Ai , Bj ))
P
(1)
n2
i,j |Ai ||Bj |
=
=
1
e2 (A, B)
P
P
n2 ( i |Ai |)( j |Bj |)
q(A, B)
Lemma 3.8. If P, P 0 are partitions of V (G) and P 0 refines P then
q(P 0 ) ≥ q(P)
.
5
k
Proof. Let
S P = {Ci }i=1 and for each Ci ∈ P let Ci be the refined partition
0
so P = Ci , then
X
q(P) =
q(Ci, Cj )
i<j
≤
(3.7)
X
i<j
≤
q(Ci , Cj )
q(P 0 )
The last inequality comes from the fact that q(P 0 ) =
(i.e. we must also measue the ”inside” of each Ci ).
P
q(Ci ) +
P
i<j
q(Ci , Cj )
We have shown that the value of q will not decrease when we refine a
partition. Now, we will show that the value of q will increase by a small
amount if we subpartition the irregular pairs in the partition.
Lemma 3.9. Let ε > 0 and let A, B ⊂ V (G) be disjoint. If (A, B) is not
ε-regular, then there exist partitions A = {A1 , A2 } of A and B = {B1 , B2 }
of B such that
|A||B|
q(A, B) ≥ q(A, B) + ε4
n2
Proof. Since (A, B) is not ε-regular, there must be sets A1 ⊂ A and B1 ⊂ B
where |A1 | > ε|A| and |B1 | > ε|B| such that
|d(A, B) − d(A1 , B1 )| > ε
(2)
Let A2 = A \ A1 and B2 = B \ B2 . Take A = {A1 , A2 } and B = {B1 , B2 } as
partitions of A and B.
We will now show that A and B satisfy the lemma.
q(A, B)
1 X e2 (Ai , Bj )
n2 i,j |Ai ||Bj |
X e2 (Ai , Bj ) 1 e2 (A1 , B1 )
+
=
n2 |A1 ||B1 |
|Ai ||Bj |
i+j>2
≥
1 e2 (A1 , B1 ) e(A, B) − e(A1 , B1 )2
(1)
+
n2 |A1 ||B1 |
|A||B| − |A1 ||B1 |
=
To shorten the notation we write ζ := d(A1 , B1 ) − d(A, B) and we get
|A1 ||B1 |
e(A, B) + ζ|A1||B1 | = e(A1 , B1 )
|A||B|
6
so
2
1
|A1 ||B1 |e(A, B)
+ ζ|A1 ||B1|
n q(A, B) ≥
|A1 ||B1 |
|A||B|
2
|A||B| − |A1 ||B1 |
1
e(A, B) − ζ|A1 ||B1 |
+
|A||B| − |A1 ||B1 |
|A||B|
|A1 ||B1 | 2
2ζe(A, B)|A1 ||B1 |
=
e (A, B) +
+ ζ 2 |A1 ||B1 |
2
2
|A| |B|
|A||B|
|A||B| − |A1 ||B1 | 2
2ζe(A, B)|A1||B1 |
+
e (A, B) −
2
2
|A| |B|
|A||B|
2
2
2
ζ |A1 | |B1 |
+
|A||B| − |A1 ||B1 |
2
e (A, B)
≥
+ ζ 2|A1 ||A1|
|A||B|
≥
e2 (A, B)
+ ε4 |A||B|
(2)
|A||B|
2
since |A1 | ≥ ε|A| and |B1 | ≥ ε|B|.
The main lemma in the proof of the regularity lemma states that if a partition, P, has too many irregular pairs to satisfy the definition of ε-regularity,
then a subpartitioning of all the irregular pairs give an increase of q(P) by
a constant and the size of the exceptional set will only increase by a little.
This will also give an upper bound on the partition size.
Lemma 3.10. Let 0 < ε ≤ 1/4 and let P = {Ai }ki=0 be a (k, c)-equitable
partition of V (G), where |A0 | ≤ εn. If P is not ε-regular then there is a
(l, c0 )-equitable partition P 0 = {Bi }li=0 of V (G) where k ≤ l ≤ k4k ,
|A00 | ≤ |A0 | +
n
2k
and
q(P 0 ) ≥ q(P) +
ε5
2
Proof. Let Aij be a partition of Ai such that if the pair (Ai , Aj ) is ε-regular
we let Aij := {Ai } and if it not, then we use lemma 3.9 to get a partition
Aij of Ai and Aji of Aj . This gives new partitions of size 2 and we get
q(Aij , Aji) ≥ q(Ai , Aj ) + ε4
7
|Ai ||Aj |
n2
Take Ai as the minimal partition of Ai that refines every partition Aij , i 6= j,
A0 = {{v} : v ∈ A0 } and let A be the partition
[
A = {A0 } Ai
i
of V (G). This is a refinement of P. Since we have |P| = k + 1 and we
disregard the exceptional set A0 , we get |Ai | ≤ 2k−1 and therefore k ≤ |A| ≤
k2k . Since P is not ε-regular, there must be more than εk 2 of the pairs
(Ai , Aj ) that are ε-irregular and hence as many Aij that uses lemma 3.9 to
refine the irregular pairs. Thus, we get by lemma 3.7 and 3.9
X
X
X
q(A) =
q(Ai , Aj ) +
q(A0 , Ai ) +
q(Ai )
i<j
≥
≥
X
i<j
X
i
1≤i
q(Aij , Aji) +
q(Ai , Aj ) +
i<j
= q(P) + ε5
≥ q(P) +
X
q(A0, {Ai }) + q(A0 )
1≤i
ε 4 c2
εk 2 2
n
(kc)2
n2
+
X
1≤i
q(A0 , {Ai }) + q(A0)
ε5
2
where we get the last inequality from the fact that |A0 | ≤ εn ≤ 41 n which
imply that kc ≥ 34 n.
Finally, we must show that all parts Ai ∈ A, i ≥ 1 can have the same size
c0 without the exeptional set growing too large from collecting the remaining
vertices. Let B1 , . . . , Bl be disjoint sets of size d = b 4ck c, chosen as large
as possible,
S such that each
S Bi is a subset of some A ∈ A \ {A0 } and A0 =
V (G) \ Bi . Take P 0 = Bi , which is a partition of V (G). Lemma 3.8 gives
q(P 0 ) ≥ q(A) ≥ q(P) +
ε5
2
Since d = b 4ck c, we have that the number of sets Bi that is a subset of some
Aj is less than 4k and therefore k ≤ l ≤ k4k as was given in the lemma.
All that remains to be shown is that the exceptional set B0 is not too
large. The sets (Bi )li=1 contains all but at most d vertices from each set
A ∈ A \ {A0 } (since we chose the collection (Bi )l1 as large as possible). We
8
have
|B0 | ≤ |A0 | + d|A|
c
≤ |A0 | + k k2k
4
ck
= |A0 | + k
2
n
≤ |A0 | + k
2
as required.
We are now ready to prove the regularity lemma. The main idea is to
iterate the procedure in lemma 3.9.
Proof of Theorem 3.4. Let 0 < ε ≤ 1/4 and m ∈ N be given. Observe that
q(P) =
X |Ai ||Aj |
n2
i<j
≤
d2 (Ai , Aj )
1 X
|Ai ||Aj |
n2 i<j
≤ 1
This gives u = 2/ε5 as an upper bound on the number of times the procedure of lemma 3.9 can be applied to an initial (irregular) partition before it
becomes a regular partition.
Now, we must show that the exceptional set A0 of the initial partition
{Ai }ki=0 does not grow too large, i.e. we must show that |A0 | ≤ εn. Lemma
3.9 gives an upper bound of n/2k on the growth of |A0 | for each iteration.
Thus, for any initial value of |A0 | ≤ k, we must choose k ≥ m large enough
such that
u
k + k n ≤ εn
(3)
2
Choose k such that 2k−1 ≥ u/ε. Then the inequality is true for all n ≥ 2k/ε.
The upper bound on the number of parts in the partition, M, is the
number we end up with after u iterations of lemma 3.9. If n ≤ M the
lemma is trivial since we may partition the vertex set into a set of singletons,
so we may assume that n > M. Lemma 3.9 gives that the partition size
(disregarding the exceptional set) grows to at most k4k for each iteration.
Let f (x) = x4x , then
2k
M = max{f ◦n (k), }
ε
where the last term is to be sure that (3) is satisfied.
9
To show that every graph of order at least m has an ε-regular partition,
{Vi }ki=1 , with m ≤ k ≤ M, we simply have to construct the partition in
the way already mentioned. Let A0 ⊂ V (G) be the minimum set such that
k | |V (G) \ A0 | and {A1 , . . . , Ak0 } be any partition of V (G) \ A0 such that
all the sets have equal size. Then |A0 | ≤ k 0 . Now we iterate the procedure
of lemma 3.9 until the partition of V (G) is ε-regular. As shown above, this
will require at most 2/ε5 iterations and the size of the exceptional set will
stay below εn.
3.4
Bounds
The upper bound on M is very large. From the proof it is possible to derive
that M is proportional to a tower of 2s of height ε−5 (see [22]), i.e. for some
constant c,

..2 
.
ε−5
M ≤ c22

In [15] Gowers proves that there exists graphs for which the size of the partition must be as large as a tower of 2s of size c0 ε−1/16 , where c0 is an arbitrary
constant.
The lower bound m is used to ensure that almost all edges occurs between
the regular pairs (if m is too small many edges occur within the partition
classes).
Even though the lemma is valid for all graphs larger than the lower bound
m, it does not always give useful results. If n < M, it is always possible to
construct the partition out of singletons and the lemma therefore give no
useful information. Hence, the large upper bound on M tell us that the
graphs must have many vertices, since we want the lemma to give a nontrivial partition of the graph. Most applications of the lemma have, in fact,
the form Given some parameters, there exists an N such that all graphs with
more than N vertices have a certain property. Since N can be very large, the
lemma is most useful to prove asymptotic results.
Most applications of the regularity lemma use the regular pairs that have
positive density, and therefore it is important that the graphs are dense, i.e.
for some fixed constant δ > 0 we have that
e(G)
>δ
n(G)
2
which is equivalent to saying that the graph has more than cn2 edges for
some fixed constant c (more on this in Section 5). If the graphs are sparse
(i.e. have o(n2 ) edges) the density will tend to 0 for all pairs.
10
Another interesting thing is the number of irregular pairs. By the definition of ε-regularity, the partition is allowed to have at most εk 2 irregular
pairs. For a long time it was not known if this was necessary at all. There
are, however, certain graphs that must have at least ck irregular pairs[2],
where c = c(ε). A simple example is the half-graph, i.e. the bipartite graph
H with partition sets A = {a1 , . . . , an }, B = {b1 , . . . , bn } and ai bj ∈ E(H)
iff i ≤ j.
Figure 1: The half-graph for n = 4.
11
4
Historical background
As I said in the introduction, the regularity lemma was not originally intended
as a powerful tool in extremal graph theory, but rather as an auxiliary lemma
to proof a theorem in additive number theory. The section will give a short
background to this area.
4.1
van der Waerden’s theorem
A classic result in combinatorial number theory is van der Waerden’s theorem, which states a Ramsey like property for the natural numbers. If we
partition the natural numbers into finitely many classes (or colours as they
are usually called in Ramsey theory), then at least one of the classes contains
an arbitrarily long arithmetic progression2 .
Theorem 4.1 (van der Waerden 1927 [40]). Let k, t ∈ N. If we colour N
in t colours then there is a monochromatic arithmetic progression of k terms.
By monochromatic we mean that all elements of the set have the same
colour and an arithmetic progression of k terms is simply a set of the form
k−1
{a + it}i=0
= {a, a + t, a + 2t, . . . , a + (k − 1)t} ⊂ Z
An equivalent finite version of the theorem is the following:
Theorem 4.2 (van der Waerden, finite version). Let k, t ∈ N then there
exists a N = N(k, t) ∈ N such that if we colour [N] with t colours, then at
least one colour-class will contain an k-term arithmetic progression.
An elegant (modern) proof of the theorem can be found in [35]. An
interesting question is how fast N(k, t) grows. The original proof by van der
Waerden gives an upper bound on M by an Ackermann-type function, even
when t = 2. Recently, this bound has been improved significantly by Gowers
[16] but it is still huge.
4.2
Szemerédi’s theorem
In 1936, Erdös and Turán conjectured a strengthening of van der Waerdens
theorem. They belived that it is possible to find arithmetic progressions of
length k in any sufficiently dense subset of Z. In 1954 K.F. Roth proved the
2
This result was first conjectured by Schur, but it is sometimes called Baudet’s conjecture since it was from Baudet van der Waerden heard of the conjecture [17].
12
conjecture for k = 3 (see section 6 for a simple proof) and in 1969 Szemerédi
proved the conjecture for k = 4 [33]. It was a major breakthrough when
Szemerédi proved the conjecture for arbitrary k:
Theorem 4.3 (Szemerédi 1975 [34]). For every k ∈ N and δ > 0 there
exists a N = N(k, δ) such that, for every n ≥ N and A ⊂ [n] with |A| > δn,
we have that A must contain an k-term arithmetic progression.
In analogy with van der Waerden’s theorem, there also exists an infinite
version of this theorem. We begin by defining the term upper density (or
Banach density).
Definition 4.4 (Upper density). The upper density of a set A ⊂ Z is
defined as
|A ∩ [−n, n]|
δ = lim sup
|[−n, n]|
n→∞
and if δ > 0 we say that A has positive upper density.
Theorem 4.5 (Szemerédi, infinite version). If A ⊂ Z has positive upper
density, then for any k ∈ N, A contains infinitly many arithmetic progressions of length k.
Another more general way to think about this is that once the density
exceeds a certain threshold certain patterns must emerge [1], or as the famous
phrase for describing Ramsey theory goes: total disorder is impossible.
There are many (very) different proof of this theorem. The original proof
was combinatorial and a key element was the main theorem of this paper,
namely the regularity lemma. A different proof was given by Fürstenberg
in 1977 [13] using ergodic theory3 . A recent proof by W. T. Gowers [16]
uses Fourier analysis and combinatorics. Gowers also proved [14] a stronger
theorem (the multidimensional Szemerédi’s theorem) using hypergraphs (see
Section 8).
The main difficulty that arises in the proof of this theorem is that one
has no a priori information of A except a lower bound on its density. The
way of dealing with this is to separate the high and low order information
of the structure of A. In his abstract of [36] Tao says that all the different
proofs of Szemerédi’s theorem . . . are based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to
a decomposition of any object into a structured (low-complexity) component
and a random (discorrelated) component.
3
Ergodic theory is the study of ergodic transformations, i.e. measure-preserving transformations on a probability space.
13
4.3
The original lemma
The lemma that Szemerédi originally used to prove his theorem [34] was a
weaker lemma formulated in far less appealing way and also only for bipartite
graphs.
Theorem 4.6 (The old regularity lemma). For every ε1 , ε2 , δ, ρ, σ ≥
0 there exists n0 , m0 , N, M ∈ N, such that for every bipartite graph G =
(A, B, E) where |A| = n ≥ N and |B| = m ≥ M there exist sets Vi ⊂ A, i <
n0 and Vij ⊂ B, j < m0 for which
[
Vi | < ρn
|A \
i<n0
|B \
[
j<m0
i<n0
Vij | < σm
and for every i < n0 , j < m0 , T ⊂ Vi , S ⊂ Vij with |T | > ε1 |Vi | and
|S| > ε2 |Vij | then
d(T, S) > d(Vi , Vij ) − δ
and for each u ∈ Vij
|N(u) ∩ Vi | < (d(Vi , Vij ) + δ)|Vi |
This version is only mentioned as a background and is seldom used today.
There are however many applications of the regularity lemma where only one
ε-regular pair is used (i.e. only a bipartite graph).
14
5
An extremal problem
5.1
The regularity graph
The (original) regularity lemma is usually applied in the context of dense
extremal graph theory. The most common application is to prove that if a
graph G is dense enough, then it must contain a given graph H as a subgraph.
It is easy to show that if G is a random graph, with n = n(G) and positive
edge density, then it almost certainly contains H as a subgraph4 when n → ∞
[4]. Since all the pairs in the ε-regular partition of a graph have uniformly
distributed edges, they behave in much the same way as random bipartite
graphs with equal partition sets. Therefore, if the parts are very large and
have a positive edge density, they contain any given bipartite subgraph.
Consider H as a union of bipartite graphs. If G is dense, then a substantial
number of the pairs must have a positive edge density, and if this number is
large enough, then it is very likely that H ⊂ G. An important concept is the
regularity-graph 5 [8].
Definition 5.1 (Regularity graph). Let R = RG (ε, l, d) be the graph with
vertex set P = {V0 , V1 , . . . , Vk }, where P is an ε-regular partition of G and
|V1 | = · · · = |Vk | = l. The edge Vi Vj ∈ E(R) if and only if the pair (Vi , Vj )
is ε-regular and d(Vi , Vj ) ≥ d. We call this graph the regularity graph of G.
s
Furthermore, let Rs = RG
(ε, l, d) be the graph where each vertex Vi ∈ V (R)
is replaced by a set Vis of s independent vertices and for any vertex u ∈ Vis
and v ∈ Vjs , uv ∈ E(Rs ) if and only if Vi Vj ∈ E(R). In other words, R is
replaced by copies of Ks,s . Rs is sometimes called the blowup graph of R.
Remark. Note that the regularity graph is not unique and is not necessary
defined for all parameters 0 < ε < 1, l ∈ N and 0 ≤ d ≤ 1.
The most common way to prove that G has a certain subgraph H using
the regularity lemma, is to first apply the lemma (with some appropriate
parameters) in order to get an ε-regular partition P and then use some other
theorem from extremal graph theory to check whether the regularity graph,
s
RG (with respect to P ) or the blowup regularity graph, RG
contains the given
subgraph. If, for instance, RG contains a triangle, then G will most likely
contain many triangles.
The following lemma formalizes the above reasoning.
4
This is rather intuitive, since we can build H vertex by vertex, and always find a new
vertex in G with the desired connections since G is large.
5
This graph is also sometime called reduced graph[24], skeleton[5] or cluster graph.
15
Lemma 5.2 (Embedding lemma). Given a graph G, d ∈ (0, 1], s ∈ N
and ∆ ≥ 1, there exists an ε0 > 0 such that if H is a graph with ∆(H) ≤ ∆
and RG (ε, l, d) is any regularity graph of G with ε ≤ ε0 and
l≥
2s
d∆
then
s
H ⊂ RG
(ε, l, d) ⇒ H ⊂ G
In order to prove the lemma, we note the following simple fact: If (A, B)
is an ε-regular pair, then for any large enough Y ⊂ B, most vertices in A
have roughly the expected number of neighbours in Y .
Lemma 5.3 (Most degrees into a large set are large). Let (A, B) be
an ε-regular pair with density d. Then for any Y ⊂ B, with |Y | > ε|B| we
have
|{v ∈ A : deg(v, Y ) ≤ (d − ε)|Y |}| ≤ ε|A|
Proof. Let X = {x ∈ A : deg(x, Y ) < (d − ε)|Y |}. Then e(X, Y ) <
|X||Y |(d − ε) and therefore
d(X, Y ) =
e(X, Y )
< d − ε = d(A, B) − ε
|X||Y |
and since |Y | ≥ ε|B| we have |X| < ε|A|.
Proof of Lemma 5.2. Let d and ∆ be given. Choose ε0 small enough that
0 < ε0 < d and
1
(4)
(d − ε0 )∆ − ∆ε0 ≥ d∆
2
Let P = {Vi }ki=0 be the ε-regular, (k, l)-equitable partition of G that gives
gives the regularity graph RG , then ε ≤ ε0 , V (R) = {V1 , . . . , Vk } and
l≥
2s
d∆
Assume that H is a subgraph of Rs , then for each ui ∈ V (H) we have that
ui ∈ Vjs ⊂ V (Rs ) for some j. This defines a map σ : i → j. We want to
show that there is an embedding ui → vi ∈ Vσ(i) such that vi vj ∈ E(G) if
and only if ui uj ∈ E(H). This can be achieved trough choosing the vertices
vi inductively. Let Yi ⊂ Vσ(i) be a set assigned to each vertex ui which
contains the possible choices for vi . Initially, let Yi = Vσ(i) . For each vertex
ui ∈ V (H), we choose a vertex vj , j < i such that when uj ui ∈ E(H), we
16
delete {vk ∈ Yi : vk vj 6∈ E(H)} from Yi . Our aim is to get a sequence of sets
Yi,0 , . . . , Yi,m such that
Vσ(i) = Yi,0 ⊃ · · · ⊃ Yi,m = {vi }
where Yi,j+1 = Yi,j ∩N(vj ). We must make sure that the sets does not get too
small. By lemma 5.3 we can choose the sets such that |Yi,j−1|−|Yi,j | < (d−)
if |Yi,j−1| < εl and all but at most εl choices of vj will satisfy
|Yi,j | ≥ (d − ε)|Yi,j−1|
(5)
Since there are at most ∆ choices of vi in total and all but at most ∆εl of
those choices satisfy (5) for all i. It remains to show that |Yi,j | > εl for all
j < m and that we have at least s ways to choose vi ∈ Yi,m−1 .
We know that |Yi,0| = |Vσ(i) | = l and hence for all j < m,
|Yi,j | − ∆εl ≥ (d − ε)∆ l − ∆εl ≥ (d − ε0 )∆ l − ∆ε0 l ≥(5)
1 ∆
d l≥s
2
This also implies that |Yi,j | ≥ εl and |Yi,j | − ∆εl ≥ s.
There are many many other stronger versions of lemma 5.2. One is the
Key lemma[24]. Let kH → Gk denote the number of labelled copies of H in
G.
Lemma 5.4 (Key Lemma). Given d > ε > 0, m ∈ N and a graph R.
Construct a graph G by replacing every vertex v ∈ V (R) by m vertices,
and replacing every uv ∈ E(R) with the ε-regular pair (us , v s ), such that
d(us , v s ) ≥ d. Let H ⊂ Rs with h vertices, ∆ = ∆(H) > 0 and let δ = d − ε
and
δ∆
ε0 =
2+∆
If ε ≤ ε0 and s − 1 ≤ ε0 m, then
kH → Gk > (ε0 m)h
Lemmata of this sort where the frequencies of certain subgraphs (up to
isomorphism) are given, are called counting lemmas. The combined use of
counting lemmas and the regularity lemma are called the regularity method
[29]. The proof of the key lemma is algorithmic and can be found in [24].
17
5.2
Erdös-Stone theorem
One of the classic results in extremal graph theory is Turán’s theorem (see
Appendix B), which says that
ex(n, Kr ) = tr−1 (n)
In 1946, Erdös and Stone gave a famous generalization of this theorem.
Theorem 5.5 (Erdös and Stone 1946 [9]). Let r, s ∈ N, r ≥ 2 and δ > 0.
Then there exists an N ∈ N such that every graph, G, with n ≥ N vertices
contains Ksr as a subgraph if
e(G) ≥ tr−1 (n) + δn2
The original proof of this theorem did not use the regularity lemma (see
e.g. [4] for a classic proof). However, it is both instructive and natural to
prove it using Turán’s theorem and the regularity lemma.
The theorem also has a very interesting corollary which states that the
maximum edge density in a n order graph not containing H as a subgraph
only depends on the chromatic number of H when n → ∞.
Corollary 5.6. Given a graph H with e(H) ≥ 1, we have
ex(n, H)
χ(H) − 2
=
n
n→∞
χ(H) − 1
2
lim
A proof of this corollary can be found in [8]. The basic idea in the proof
of theorem 5.5 is to use the regularity lemma on a graph G to create the
regularity graph RG and to show that it is dense enough that Kr ⊂ RG
s
and lemma 5.2 therefore tells us that
by Turán’s theorem. Then Krs ⊂ RG
s
Kr ⊂ G.
Proof of theorem 5.5. Let δ > 0, r ≥ 2 and s ≥ 1 be given. The result
follows directly from Turán’s theorem when s = 1, so we can assume that
s ≥ 2. Let G be any graph with n = n(G) vertices and
e(G) ≥ tr−1 (n) + δn2
edges. Since G is simple we get δ < 1.
Let d = δ and ∆ = ∆(Ksr ). Lemma 5.2 then gives an ε0 . Apply the
regularity lemma with 0 < ε ≤ ε0 ,
ε<
δ
<1
2
18
(6)
m>
1
δ
and
1
>0
(7)
m
The regularity lemma returns an upper bound on the size of the partition
M. Assume that
2Ms
(8)
n≥ ∆
d (1 − ε)
γ := 2δ − ε2 − 4ε −
Let P = {Vi }ki=0 be an (k, l)-equitable ε-regular partition of G, where m ≤
k ≤ M. Since the exceptional set V0 does not need to be empty, we get
n ≥ kl
(9)
and
n − εn
2s
n − |V0 |
≥
≥(8) ∆
k
M
d
r
Now lemma 5.2 imply Ks ⊂ G if Kr ⊂ RG (ε, l, d). Thus, all we need to
show is that RG is dense enough to contain Kr , i.e. d(Vi , Vj ) ≥ d for enough
regular pairs (Vi , Vj ).
We know that
1
|V0 |
≤ (εn)2
e(V0 ) ≤
2
2
l=
and
k
X
i=1
e(V0 , Vi ) ≤ |V0 |kl ≤ εnkl
Furthermore there are at most εk 2 irregular pairs, each containing at most l2
edges. By the definition of density we get that there are less then dl2 edges
in the ε-regular pairs with density < d. Each Vi , 1 ≤ i ≤ k has at most 2l
edges. All other edges contribute to the edges in RG . Hence
l
1
2
2 2
2
e(G) ≤ (εn) + εk l + dl +
k + e(RG )l2
2
2
From Appendix B we know that
1 r−2
=
tr−1 (n) ≤ n2
2 r−1
19
1−
2
1
n
r−1 2
(10)
For sufficiently large n this imply
e(R)
≥
≥(6),(9)
≥(9)
=(7)
>lemmaB.5
≥(10)
l
1 2 e(G) − 21 (εn)2 − εk 2 l2 − dl2 − 2 k
k
1 2 2
2
k l
2
2
1 2 tr−1 (n) + δn − 21 ε2 n2 − εnkl
1
− 2ε − d −
k
1 2
2
k
n
2
1 2 tr−1 (n)
1
k
+ 2δ − ε2 − 4ε − d −
1 2
2
m
n
2
−1 1
n
1 2
1−
k tr−1 (n)
+γ
2
2
n
1 2r − 2
k
2 r−1
tr−1 (k)
By Turán’s theorem, we get Kr ⊂ RG .
In fact, by using the Key lemma in a similar way, it is possible to prove
the following stronger theorem
Theorem 5.7 (Number of copies of H [24]). Let H be a graph of order
h, β > 0 and choose ε = (β/6)h . Now there exists an N ∈ N such that for
all graphs G with n vertices where n > N and
2
n
1
+β
e(G) > 1 −
χ(H) − 1
2
we have
kH → Gk >
εn
M(ε)
h
where M(ε) is the upper bound on the partition size in the regularity lemma.
20
6
3-term arithmetic progressions
The simplest non-trivial case of Erdös-Turán’s conjecture is when k = 3
which was proven by Roth in 1954 (see Section 4.2). This result can also
be proven using the regularity lemma (without giving the full proof of Szemerédi’s theorem). In this section we will give this proof and by doing this
also illustrate a way of using the regularity lemma that differ from the one
in the previous section. This also gives an idea of how the lemma is used in
additive number theory.
6.1
Induced matching problem
We begin with a related, but slightly different problem: How many edges can
a graph that is a union of induced matchings have?
Theorem 6.1. Let G be a graph with n vertices, then if E(G) is the union
of induced matchings6, we have
e(G) = o(n2 )
Proof. Assume that e(G) > cn2 for some fixed c > 0 (i.e. G is dense). Apply
the regularity lemma with ε < c/8 and m large enough that there are at
least cn2 /4 edges between the ε-regular pairs. We get P = {Vi }ki=0 as the
ε-regular partition.
Denote the matchings composing E(G) by M1 , . . . , Mn and let
c 2
0
G = G \ Mi : e(Mi ) < n
2
Since at most cn2 /2 edges are deleted, we have e(G0 ) ≥ cn2 /4.
If |V (Mj ) ∩ Vi | ≤ c|Vi|/8 for some 1 ≤ i ≤ k, 1 ≤ j ≤ n we let Mj0 be the
matching we get by deleting all edges from Mj that are incident to Vi . Let
00
G =
n
[
Mi0
i=1
Then
e(G00 ) ≥
cn2
8
P
since at most ki=1 c|Vi|/8 = cn/8 edges are deleted in each Mj0 . For each
edge in G00 let (Vi , Vj ) be the ε-regular pair that contains it and Ml0 be the
6
See Appendix A
21
matching. We let A = Vi ∩ V (Ml0 ) and B = Vj ∩ V (Ml0 ) and obviously we
have |A| ≥ 8c |Vi | and |B| ≥ 8c |Vj |.
Since A and B are both in the induced matching Ml0 we have
e(A, B) ≤ min(|A|, |B|)
Assume that |A| is the smallest, then
d(A, B) ≤
1
8
min(|A|, |B|)
≤
≤
|A||B|
|B|
c|Vj |
The regularity lemma tells us that |Vj | can become arbitrary large when n
grows (k has an upper bound M(ε, m)) and since the pair (Vi , Vj ) is ε-regular,
we have |d(Vi , Vj ) − d(A, B)| < ε. This imply that |d(Vi, Vj )| ≤ 2ε and hence
X
e(G00 ) ≤ |{vi vj ∈ V (G00 ) : (Vi , Vj ) is irregular}| +
e(Vi , Vj )
(Vi ,Vj ) ε-regular
≤ εn2 + 2ε
X
1≤i<j≤k
|Vi ||Vj | < 3εn2
Since e(G00 ) ≥ cn2 /8, c must be arbitrarily small, but this contradict the
assumption.
Remark. By throwing away insignificant parts of the graph, we have much
better control over the remaining parts. This is a very common thing to
do when applying the regularity lemma and it is known as the Hungarian
method.
6.2
The (6, 3) problem
A classic problem in extremal (hyper)graph theory is the (6, 3)-problem:
which is the maximum number of hyper edges a 3-uniform7 hypergraph can
have such that no 6 vertices have 3 or more hyperedges (triangles) between
them. This is a simple (although far from trivial) special case of a more
general question asked by Brown, Erdös and Sós. The problem was solved
by Ruzsa and Szemerédi.
Theorem 6.2 (Ruzsa-Szemerédi 1976 [30]). Let H be a 3-uniform hypergraph on n vertices. If there are no 6 vertices with 3 or more hyperedges
between them, then
e(H) = o(n2 )
7
All edges are 3-subsets of the vertex set (see Appendix A).
22
Proof. Let H be the 3-uniform hypergraph from the theorem. Define the
ordinary graph G over the same vertex set and let uv ∈ E(G) iff {u, v} ⊂
e ∈ E(H). For some v ∈ V (G) with deg(v) ≥ 3, let
Mv = {e \ {v} : e ∈ E(H) and v ∈ e}
and if deg(v) ≤ 2 we let Mv be empty.
Now Mv forms a matching of G since two hyperedges only can intersect
at v (see figure 2) and this imply that Mv is disjoint.
Figure 2: The following configuration is the result of a vertex v with deg(v) ≥
3 has edges that also intersect at another vertex than v. This violates the
(6, 3)-condition of the theorem.
It is also easy to show that the matching is induced. Assume that it is
not induced. Then there must be an edge in G that connects two different
parts of the matching, and we have the situation in figure 3.
Figure 3: If the matching is not induced, there must be an edge between
different parts of the matching and since all edges comes from triangles, this
violates the (6, 3)-condition.
23
If G0 is the graph formed by taking the union of all matchings, then from
Theorem 6.1 we know that
e(G0 ) = o(n2 )
Since the edges in G0 are formed from the edges in H and the only edges that
does not contribute to edges in G0 are those adjacent to a vertex of degree
at most 2 in G, we have
e(H) ≤ e(G0 ) + 2n
and hence
e(H) = o(n2 )
6.3
3-term arithmetic progressions
In order to prove Roth’s theorem, we begin by proving the following stronger
theorem.
Theorem 6.3 (Ajtai-Szemerédi 1974). For any δ > 0 there exists N0 ∈ N
such that if N > N0 every S ⊂ N2 with |S| ≥ δN 2 contains a triple of the
form {(a, b), (a + d, b), (a, b + d)} for some a, b, d ∈ N and d 6= 0.
The followning proof is due to Solymosi [32].
Proof. Let S ⊂ N2 such that |S| ≥ δN 2 . Define a bipartite graph G with
bipartitions A = {a1 , . . . , aN } and B = {b1 , . . . , bN }. Let ai aj ∈ E(G) iff
(i, j) ∈ S.
Partition the edges in G into equivalence classes ai bj ∼ ak bl iff i+j = k+l.
Since each class is a matching, we can use Theorem 6.1 and if N is sufficiently
large we know that at least one matching is not induced. Hence, a triple
of edges ai bl , ai bj , ak bl such that ai bj ∼ ak bl guarantees that {(a, b), (a +
d, b), (a, b + d)} ⊂ S.
It is rather easy to prove that Ajtai-Szemerédi’s theorem imply Szemerédi’s theorem for k = 3 (Roth’s theorem).
Theorem 6.4. For every δ > 0 there exists an N0 = N0 (δ) such that for
all N > N0 we have that, if A ⊂ [N] with |A| > δN, A contains a 3-term
arithmetic progression.
Proof. Define S = {(a, b) : a − b ∈ A and a, b ∈ [N]}. Since |S| ≥ 21 |A|2 >
1
(δn)2 we can apply Theorem 6.3 with 21 δ 2 . If N is sufficiently large we know
2
that {(a, b), (a + d, b), (a, b + d)} ⊂ S and hence, {a − b − d, a − b, a − b + d} ⊂
A.
24
A natural question is whether similar reasoning can be used to prove Szemerédi’s theorem for 4-term arithmetic progressions or perhaps even n-term
arithmetic progressions. The approach that we have seen uses mainly properties of pairs and in the case of 4-term arithmetic progressions we would
have to use triplets. The main difficulty that arises here is to extend the regularity lemma to 3-uniform hypergraphs (where each hyperedge is a triple of
vertices) in a useful way. In section 8.2 a hypergraph version of the regularity
lemma is presented and in [1] a proof of the 4-term arithmetic progression
can be found that uses this result.
25
7
Algorithmic aspects of the regularity lemma
The regularity lemma has numerous applications in important applied problems such as e.g. graph colourings, graph embeddings and graph decomposition. Therefore it would be of interest to find an algorithmic version of
the regularity lemma that can create the regularity partition. Although the
proof in Section 3.3 contains algorithmic parts it is mainly non-algorithmic.
However it gives us a hint of an algorithm. From Lemma 3.10 we can deduce
two things that we need. First we must be able to efficiently check whether
the partition P of V (G) is ε-regular, and secondly, if it is not regular we
must create a new partition P 0.
7.1
Testing for regularity
Is it possible to efficiently test if a given partition is ε-regular? It turns out
that this is not the case (unless P = co − NP ).
Theorem 7.1 (co-NP-completeness result [2]). Given ε > 0, k ∈ N, a
graph G and a partition P = {Vi }ki=0 of V (G), then the problem of deciding
whether or not P is ε-regular is co-NP-complete.
Here we will sketch the proof of the following stronger theorem which
implies Theorem 7.1. See Appendix D for a introduction to the class NP.
Theorem 7.2. Given ε > 0 and a bipartite graph B with vertex classes X, Y
of equal cardinality. Then the problem of determining if B is ε-regular is
co-NP-complete.
This is the same as to determine if P = {∅, X, Y } is an ε-regular partition
of B.
Proof sketch. Our goal is to show that the complement of the problem is
NP-complete. The main idea is a reduction from the known NP-complete
problem CLIQUE, i.e. the problem of determine if a graph has a clique of a
given size. This problem can be reduced to the Kk,k problem, which is the
problem of testing if Kk,k is a subgraph of a bipartite graph G with vertex
classes of size n. This is then used to prove the following lemma.
Lemma 7.3. Given a bipartite graph B with partition classes X, Y , where
2
|X| = |Y | = n and e(B) = n2 − 1, the problem of deciding if B contains a
subgraph isomorphic to K n2 , n2 is NP-complete.
26
Let B be a bipartite graph with vertex classes X, Y of size n and e(B) =
− 1. Then the claim is that K n2 , n2 ⊂ B iff B is not ε-regular for ε = 21 . To
prove the claim, we first assume that B is not ε-regular. Then there must
exists a pair (C, D), where C ⊂ X and D ⊂ Y and |C|, |D| ≥ 21 n, that is a
witness to the irregulariy. Thus
1
1
1
|d(C, D) − d(X, Y )| = d(C, D) − + 2 ≥
2 n
2
n2
2
and therefore d(C, D) = 1. This implies that K n2 , n2 ⊂ B[X ∪ Y ]. The other
direction is trivial since B is not allowed to have any irregular pair (because
k = 1) and if we choose C and D to be exactly those vertices that contains
K n2 , n2 then we have an irregular pair.
This reduction can be done in linear time and hence the problem is coNP-complete.
See [2] for a complete proof.
At first this may seem to impose a major problem to the approach described in the beginning of this section. However, it turns out that we only
need to solve a weaker problem in order to create the regularity partitions.
Lemma 7.4 ([21, 23]). Given a bipartite graph B with partition classes of
size n and a real number ε > 0 there exists polynomial time algorithm A
and a function ε0A : R+ → R+ such that A either correctly asserts that B is
ε-regular, or else gives a witness for the ε0A (ε)-irregularity.
In [23] it is shown that the complexity of A is O(n2 ) when ε0A (ε) =
ε20 /54 232 . A similar result is given in [2] where the complexity is O(M(n)) =
O(22.376 ), M(n) being the time8 required to square an n × n matrix over
{0, 1}.
A closer description of A which uses linear-sized expanders can be found
in [23].
This lemma leaves open how A behaves when B is ε-regular but not ε0 regular. Despite this fact, and the weakness that ε0 ε, the lemma does
imply that there exists a polynomial time algorithm that can create the εregular partition of a given graph.
7.2
Creating the regularity partitions
What remains in order to create the regularity partition is to create the
refined partition P 0 in Lemma 3.10. In [2] it is shown that this can be done
in linear time by the following lemma.
8
It is however notoriously hard to implement an algorithm that can achieve this.
27
Lemma 7.5 ([2]). Let k ∈ N, 0 < γ < 1 and G be a graph on n vertices. Furthermore, let P = {Vi }ki=0 be an equitable partition of V (G) and
assume that |Vi | > 42k and 4k > 600γ −5. Given witnesses that more than γk 2
pairs (Vi , Vj ) are not γ-regular, it is possible to find, in time O(n), a refined
partition P 0 into 1 + k4k classes such that
q(P 0) ≥ q(P ) +
γ5
.
20
and with an exceptional class V00 where
n
4k
From the above lemma, it is easy to derive a constructive version of the
regularity lemma.
|V00 | ≤ |V0 | +
Theorem 7.6 (Regularity lemma, constructive version [23]). Let ε >
0 and m ∈ N, then there exists M(ε, m) ∈ N such that every graph with
n > M(ε, m) vertices has an ε-regular partition P = {Vi }ki=0 where m ≤ k ≤
M(ε, m) and the partition can be found in O(n2 ) sequential time.
See [2] for a proof of the lemma. The algorithm that finds a regularity
partition can be formulated as follows.
1. Divide V (G) into an arbitrary equitable partition P1 = {V0 , . . . , Vb}
where |Vi | = bn/bc. This implies that |V0 | < b. Let k1 = b.
2. For every pair (Vi , Vj ) of Pi , verify if is ε-regular or find a witness of
the ε0A (ε)-irregularity.
3. If there are at most ε k2i pairs that are not verified as ε-regular, then
we have the desired partition and the algorithm halts.
4. Apply Lemma 7.5 where P = Pi , k = ki and γ = ε0A (ε) to get P 0 as
the refined partition with 1 + ki4ki classes.
5. Let ki+1 := ki 4ki , Pi+1 := P 0 , i := i + 1 and go to step 2.
The number of iterations the algorithm require does not depend on n and
the running time of each iteration is bounded by the running time of A and
hence the algorithm runs in time O(n2 ).
Another simple algorithm that creates the regularity partition using singular value decomposition of matrices can be found in [12]. This algorithm
is probably easier to implement than the one above.
The result can be improved significantly if randomization is allowed. In
[11] Frieze and Kannan shows that there is a randomized algorithm that can
create the regularity partition in randomized time O(n).
28
7.3
Algorithmic applications
Using Theorem 7.6 many algorithmic results can be found in applications
where the original lemma only gives existence results. The following two
theorems are examples of the usefulness of the algorithmic regularity lemma.
Theorem 7.7 ([2, 23]). Let ε > 0 and h ∈ N. Then there exists n0 ∈ N
such that for every graph H on h vertices and with chromatic number χ(H),
there are a deterministic algorithm of time complexity O(n2 ) that takes as
input a graph G on n > n0 vertices and
δ(G) >
χ(H) − 1
n
χ(H)
and then finds a set of (1 − ε) nh vertex disjoint copies of H in G.
This theorem was originally proven in [2] but the complexity was improved
in [23]. See [2] for the outline of the proof.
Theorem 7.8 ([23]). Given ε, c > 0 and an integer k ≥ 3, there exists an
integer n0 = n0 (k, ε) and a function f (k, ε) such that if G is a graph on
n > n0 vertices and cn2 edges (G is dense), then either
(i) there exists a graph G0 ⊂ G with χ(G0 ) ≥ k and |V (G0 )| ≤ f (k, ε), or
(ii) there exists a set of edges E 0 ⊂ E(G) with |E 0 | ≤ εn2 such that the
subgraph G00 = G \ E 0 have χ(G00 ) ≤ k − 1.
There also exists an algorithm of time complexity O(n2 ) which takes G as
input and either returns G0 in case (i) or the set E 0 of edges in case (ii),
together with a proper (k − 1)-colouring of G00 .
29
8
Extending the regularity lemma
There are many ways to make the regularity lemma stronger. As we have
seen, the original lemma is only useful for large dense graphs. Two obvious
strengthenings would be to extent the lemma to sparse graphs and hypergraphs. Another not so obvious approach was taken by Terence Tao in [37],
where he extents the lemma to probability theory and information theory.
8.1
The regularity lemma for sparse graphs
As we have seen earlier in this paper, the regularity lemma gives no useful
information if the graphs are sparse. It is however possible to strengthen the
lemma a bit by extending it to sparse graphs. In [22, 20] Kohayakawa and
Rödl give two different regularity lemmas for sparse graphs.
We begin with some definitions similar to those for the dense case. Given
a graph G, let P0 = {Vi }li=0 be a partition of V (G). We write (X, Y ) ≺ P0 if
X ∩ Y = ∅ and for some i 6= j (0 ≤ i, j ≤ l) we have X ⊂ Vi , Y ⊂ Vj .
Definition 8.1. Let 0 < η ≤ 1. Then G is (P0 , η)-uniform if, for all X, Y ⊂
V (G) with (X, Y ) ≺ P0 and |X|, |Y | ≥ ηn(G) there is a constant p ∈ (0, 1]
such that
|eG (X, Y ) − p|X||Y || ≤ ηp|X||Y |
Definition 8.2. Let H ⊂ G be a spanning subgraph of G. For X, Y ⊂ V (G),
let
(
eH (X,Y )
if eG (X, Y ) > 0
eG (X,Y )
dH,G (X, Y ) =
0
if eG (X, Y ) = 0
Definition 8.3. Given ε > 0 and let X, Y be disjoint subsets of V (G). We
call the pair (X, Y ) an (ε, H, G)-regular pair if for all A ⊂ X, B ⊂ Y , where
|A| ≥ ε|X|, |B| ≥ ε|Y |, we have
|dH,G (A, B) − dH,G (X, Y )| ≤ ε
Definition8.4. A k-equitable partition P = {Vi }ki=0 is (ε, H, G)-regular if
at most ε k2 pairs (Vi , Vj ) are not (ε, H, G)-regular.
If P and P 0 are two equitable partitions then, as before, we say that
P 0 refines P if every non-exceptional class of P 0 is contained in some nonexceptional class of P .
Theorem 8.5 (Regularity lemma for sparse graphs [22]). Let ε > 0
and k0 , l ∈ N. Then there exist η = η(ε, k0, l) > 0, K = K(ε, k0, l) ≥ k0 and
30
N = N(ε, k0 , l), such that for any (P, η)-uniform graph G of order n ≥ N,
where P0 = {Vi }li=0 is a partition of V (G), and H ⊂ G is a spanning subgraph
of G, there is an (ε, H, G)-regular k-equitable partition of V (G) refining P0
with k0 ≤ k ≤ K.
Remark. The lemma is applied on subgraphs of (P0 , η)-uniform graph. The
partition P0 = {Vi }li=0 is introduced to handle l-partite graphs and if G = Kn
we get the original lemma.
8.2
The hypergraph version of the regularity lemma
There are many difficulties that arises when one tries to extend the regularity
lemma to hypergraphs. The first problem is to extend the concept of εregularity in a both intuitive and useful way.
In the case of ordinary graphs one can say that the regularity lemma
regularizes the edges (2-tuples) versus the vertices (1-tuples). For a k-uniform
hypergraph there are k − 1 different versions of the regularity lemma. There
are many different formulations of a regularity lemma for hypergraph and the
simplest is probably the following formulation of Fan Chung [7]. Suppose that
G is a k-uniform hypergraph and r < k is an integer. Let S1 , S2 , . . . , S(k) be
r
V (G)
k
a partition of r into r partition classes. We define
k
e
EG (S1 , . . . , S(k) ) = e ∈ E(G) : |
∩ Si | = 1 for all i ∈
r
r
r
and eG (S1 , . . . , S(k) ) = |EG (S1 , . . . , S(k) )|. If G is a complete hypergraph (i.e.
r
r
E(G) = V (G)
)
we
simply
write
E
(S
G
1 , . . . , S(k) ) = E(S1 , . . . , S(k) ). We can
k
r
r
now define an hypergraph analogue of the density for a pair of vertex sets.
Definition 8.6 ((k, r)-density). The (k, r)-density dk,r is defined in the
following way
eG (S1 , . . . , S(k) )
r
dk,r (S1 , . . . , S(k) ) =
r
e(S1 , . . . , S(k) )
r
Remark. If k = 2 and r = 1 then d2,1 is identical to the density in definition
3.1 for ordinary graphs.
Using this, it is possible to define a hypergraph analogue of ε-regularity.
Definition 8.7 ((k, r, ε)-regularity). {S1 , . . . , S(k) )} is (k, r, ε)-regular if
r
all Ti ⊂ Si where
e(T1 , . . . , T(k) ) > εe(S1 , . . . , S(k) )
r
r
31
we have
|dk,r (S1 , . . . , S(k) ) − dk,r (T1 , . . . , T(k) )| < ε
r
r
Remark. In general the greater the value of r is, the stronger is the measurement of ε-regularity [7].
We are now ready to state a hypergraph version of the regularity lemma.
The proof can be found in [7].
Theorem 8.8 (Hypergraph regularity lemma [7]). Let r and k be integers such that 1 ≤ r ≤ k. For every ε > 0, there exists an integer t = t(ε)
such that for every k-uniform hypergraph G, it is possible to partition V (G)
r
into P = {S1 , . . . , Sl } where l < t, so that all but at most ε|V (G)|k are not
contained E(Si1 , . . . , Si k ) for some i1 , . . . , i(k) where 1 ≤ i1 ≤ · · · < i(k) ≤ l
r
r
(r )
and {Si1 , . . . , Si k } is (k, r, ε)-regular.
(r )
This version of the hypergraph regularity lemma is not strong enough to
give a simple proof of Szemerédi’s theorem. There are, however, stronger
versions by e.g. Tao and Gowers that can be used to give a relatively short
proof to the theorem.
In order for a hypergraph regularity lemma to be useful, it is often desirable to have a lemma that gives us information about the subgraph of a given
graph. If we, for instance, have a k-uniform hypergraph and try to regularize
the k-tuples versus the 1-tuples, we have no natural hypergraph analogue to
a counting lemma (such as Lemma 5.4) [29]. The stronger versions of the
hypergraph regularity lemma does luckily have corresponding lemmas of this
sort, such as Gowers’ Hypergraph removal lemma [16, 14].
32
9
9.1
Recent results using the regularity lemma
Uniform edge distribution and k-universal graphs
As we have seen, the ε-regularity property tells us how uniformly distributed
the edges are in a bipartite graph. A natural question is whether there is
a similar property for general graphs. In [22] Kohayakawa and Rödl give a
definition that captures this. This section will discuss some of their results.
Definition 9.1. Given γ, δ, σ > 0, we say that a graph G of order n has
property R(γ, δ, σ) if, for all S ⊂ V (G) with |S| ≥ γn we have that
|S|
|S|
.
≤ e(G[S]) ≤ (σ + δ)
(σ − δ)
2
2
It is easy to see the similarity between the definition of ε-regularity and
the above definition. In a way, they both capture the uniformity of the
edge-distribution in a graph.
Another similar property is the following.
Definition 9.2 (k-universal). Given a k ∈ N we say that a graph G is
k-universal if it contain all graphs on k vertices as induced subgraphs.
It can be shown that most large graphs actually has the above properties.
To make this more precise, we need some notations. We let G(n, m) denote
the set of all labelled graphs with n vertices and m edges. It is easy to see
that for all n ∈ N and 0 ≤ m ≤ n2 we have
n
2
.
|G(n, m)| =
m
Furthermore we let R(n, m; γ, δ, σ) ⊂ G(n, m) denote the graphs G ∈ G(n, m)
satisfying property R(γ, δ, σ) and U(n, m; k) ⊂ G(n, m) denotes the k-universal
graphs. It is possible to show the following fact.
Proposition 9.3 ([22]). Given
an integer k ≥ 1, real numbers 0 < γ, δ ≤ 1,
0 < σ < 1 and m(n) = bσ n2 c we have
|U(n, m(n); k)|
=1
n→∞ |G(n, m(n))|
lim
and
|R(n, m(n); γ, δ, σ)|
=1
n→∞
|G(n, m(n))|
lim
33
It is in fact also possible to prove that almost all large bipartite graphs
are ε-regular [22].
What is interesting is that the property of uniform edge distribution actually implies the property of universality. This is made formal in the theorem
below. It is also interesting to note that it is possible to strengthen the notion of k-universality to include information on the number of copies k-vertex
graps for k ≥ 4 so that the properties become equivalent (see [22]).
Theorem 9.4 (Kohayakawa and Rödl [22]). Let k ≥ 1 be an integer and
σ, δ ∈ R with 0 < σ, δ < 1 such that δ < σ < 1 − δ. Then there exists a real
number γ > 0 and an integer N0 for which every graph G with n(G) ≥ N0
that satisfies property R(γ, δ, σ) is k-universal.
This theorem has a nice application in Ramsey theory.
Corollary 9.5. For any collection of graphs H1 , . . . , Hr there is a graph G
such that however we colour the edges of G with colours from [r], there must
be some colour class i so that G contains an induced subgraph H 0 which is
isomorphic to Hi .
In order to prove Theorem 9.4 we need an embedding lemma similar to
Lemma 5.2.
Definition 9.6. A graph G has property P(k, l, β, ε) if its vertex set can be
partitioned into a partition P = {V1 , . . . , Vk } such that
(i) |V1 | = · · · = |Vk | = l
(ii) (Vi , Vj ), 1 ≤ i < j ≤ k are ε-regular
(iii) β < d(Vi, Vj ) < 1 − β for all 1 ≤ i < j ≤ k
Lemma 9.7 (Another embedding lemma). For all 0 < β < 12 and
integers k ≥ 1, there exist ε0 = ε0 (k, β) > 0 and l0 = l0 (k, β) such that for
every graph G and integer l ≥ l0 with property P(k, l, β, ε0 ) we know that G
is k-universal.
At first this may seem to be almost the same result as Lemma 5.2. This
is however not the case. There are some fundamental differences. This
embedding lemma deals with induced embeddings, where the embedding
lemma in section 5 gives us information of subgraphs of a bounded degree.
Another difference is that the density between the partition classes in the
definition of property P(k, l, β, ε) is bounded both above and below, where
the definition of a regularity graph only give a lower bound. One major
34
advantage with this lemma is that it enables us to embed larger graphs (but
the price for this is the requirement of a bounded density).
The proof of Lemma 9.7 is based on induction of k and it is not difficult.
It can be found in [22].
Proof of Theorem 9.4. Let δ and σ be as in the theorem and let δ1 = max{σ+
δ − 21 , 12 − σ + δ}. We then have
0<
1
1
1
1
− δ1 ≤ − ( − σ + δ) = σ − δ ≤ σ + δ ≤ + δ1 < 1
2
2
2
2
(11)
Hence, property R(γ, δ, σ) imply property R(γ, δ1 , 12 ). Let β = 12 −δ. Assume
that σ = 12 and k ≥ β3 . Apply Lemma 9.7 with β and k to get ε0 and l0 .
Choose
1
, ε0 }
ε = min{
R(k, k, k; 2)
where R(k, k, k; 2) is the usual Ramsey number (see Appendix C). Apply
the regularity lemma with ε and m = R(k, k, k; 2) to obtain the constant
M(ε, m). Define
1
N0 =
M(ε, m)l0
1−ε
and
k(1 − ε)
γ=
M(ε, m)
What remain now is to show that this choices for N0 and γ satisfy the conclusion of the theorem.
Suppose that a graph G with n(G) ≥ N0 satisfies property R(γ, δ, 21 ).
Let {Vi }ti=0 be the ε-regular partition of V (G) with m ≤ t ≤ M(ε, m) and
|V1 | = · · · = |Vt | = l that the regularity lemma ensures the existence of.
Furthermore, let RG (ε, l, 0) be the reduced graph of G. Then
t
e(RG (ε, l, 0)) ≥ (1 − ε)
2
Turán’s theorem (see Appendix B) now imply that RG (ε, l, 0) has a clique
with R(k, k, k; 2) vertices. Suppose that the clique is induced by the vertices
1, . . . , R(k, k, k; 2). Then all the pairs (Vi , Vj ) with 1 ≤ i < j ≤ R(k, k, k; 2)
is ε-regular.
Partition [R(k,k,k;2)]
into 3 parts, T1 , T2 , T3 so that the pair (i, j) ∈ T1
2
β
iff d(Vi , Vj ) ≤ 2 , (i, j) ∈ T2 iff β2 < d(Vi, Vj ) < 1 − β2 and (i, j) ∈ T3 iff
d(Vi , Vj ) ≥ 1 − β2 . Then there is a set J ⊂ [R(k, k, k, 2)] of cardinality k
such that RG (ε, l, 0)[J] is monochromatic (from the definition of the Ramsey
35
number), i.e. 2j ⊂ Tα for some α ∈ {1, 2, 3}. We now create a new graph
that is induced by this set in the following way.
[ 0
G =G
Vj
j∈J
Suppose that α = 1 then
2
βk 2 l2 kl2
k βl
l
1
kl
0
≤
e(G ) ≤
+k
+
<
−δ
2 2
2
2
4
2
2
and since
n(G0 ) = kl ≥
1 − εkn
= γn
M(ε, m)
we have a contradiction to property P(γ, δ, 21 ). Hence α 6= 1. If α = 3 we get
kl
1
k
β 2
0
+δ
l >
e(G ) ≥
1−
2
2
2
2
and one again we get a contradiction. Therefore α = 2. Since
l≥
(1 − ε)n
≥ l0
M(ε, m)
we know that G0 satisfies property P(k, l, β2 , ε0 ) for all l ≥ l0 . Lemma 9.7
now implies that G0 is k-universal.
9.2
The blow-up lemma
As we already have seen, there are several different embedding lemmas based
on the regularity lemma. A recent and very powerful lemma of this sort that
can embed spanning subgraphs into dense graphs is the Blow-up lemma [25].
In order to state the blow-up lemma, we first need a stronger definition
of regularity.
Definition 9.8 (Super-regular). Let G be a graph and (A, B) be a regular
pair of G. Then the pair (A, B) is (ε, δ)-super-regular if for all a ∈ A, we
have that deg(a) ≥ δ|B| and for all b ∈ B, we have that deg(b) ≥ δ|A|.
We also need a property similar to property P(k, l, β, ε) but with the
stronger requirement that all the pairs are (ε, δ)-super-regular and with the
upper bound on the density removed.
36
Definition 9.9. A graph G has property Q(k, l, β, ε, δ) if its vertex set can
be partitioned into a partition P = {V1 , . . . , Vk } such that
(i) |V1 | = · · · = |Vk | = l
(ii) All (Vi , Vj ), 1 ≤ i < j ≤ k are (ε, δ)-super-regular
(iii) d(Vi, Vj ) > β for all 1 ≤ i < j ≤ k
Theorem 9.10 (Blow-up lemma [25]). For all integers k, ∆ ≥ 1 and real
numbers β, δ > 0, there exists ε = ε(k, β, δ, ∆) > 0 and l0 = l0 (k, β, δ, ∆) such
that every graph that satisfies property Q(k, l, β, ε, δ) with l ≥ l0 contains all
graphs H with ∆(H) ≤ ∆ that admits to a k-colouring in such a way that
every colour occurs at most l times.
This formulation of the blow-up lemma is due to Kohayakawa and Rödl
[22]. A proof of the lemma can be found in [25]. There also exists a nice
algorithmic version that can be found in [26].
The blow-up lemma has recently been used to prove many hard conjectures (at least asymptotically) regarding e.g. powers of Hamiltonian cycles
such as Seymour’s conjecture and Pósa’s conjecture [27].
37
A
Graphs and hypergraphs
This appendix will only define some of the basic concepts that occur in this
paper. For a much more comprehensive introduction to graph theory see e.g.
[41], [8] and [5].
A graph can intuitively be seen as a collection of points (vertices) and
some lines between them (edges).
Definition A.1 (Graph).
A graph is a pair G = (V, E), where V is the
vertex set and E ⊆ V2 is the edge set. It is customary to denote the vertex
set of G as V (G) and the edge set as E(G). To avoid ambiguities in the
notation one may assume that V ∩ E = ∅.
Hypergraphs is a generalisation of graphs, where edges can contain more
than two vertices.
Definition A.2. A hypergraph is a pair H = (V, E) where V is the set
of
V
vertices and each e ∈ E an arbitrary subset of the vertex set. If E ⊆ k , i.e.
all edges contain k vertices, we say that H is a k-uniform hypergraph.
Definition A.3. A set of edges M is a matching in a graph G if no pair of
edges shares endpoints. We say that M is a induced matching if all edges in
G between the vertices of M are edges in M.
Definition A.4. Let S ⊂ N be a set of k colours. A vertex colouring of
a graph G is a function c : V (G) → S such that, for all u, v ∈ V (G) with
uv ∈ E(G), we have c(v) 6= c(u). The chromatic number χ(G) is the smallest
k ∈ N such that G has a proper k-colouring.
B
Extremal graph theory
Extremal graph theory study how global properties of a graph such as chromatic number, minimum vertex degree or edge density affects local structures
of the graph. The archetypical extremal graph problem is to find the maximum numbers of edges in a graph not containing another given graph as a
subgraph (this is known as Turán’s problem).
Definition B.1. A graph G 6⊃ H on n vertices is called extremal for H if
it has the largest possible number of edges. The maximum number of edges
an n-vertex graph can have without containing H as a subgraph is denoted
ex(n, H).
38
One of the questions Turán asked was which graphs on n vertices have
the maximum number of edges without containing Kp as a subgraph. He
constructed the following class of graphs.
Definition B.2 (Turán graphs). The complete (p − 1)-partite graph on
n ≥ p − 1 vertices, whose partition classes differ in size by at most 1, is
denoted by T p−1(n). This graph is called the Turán graph. The number of
edges in T p−1(n) is denoted by tp−1 (n).
It is easy to show that the Turán graphs are indeed extremal for complete
graphs.
Theorem B.3. Every graph G 6⊃ Kp with n vertices and ex(n, Kp ) edges is
a T p−1 (n).
Proof. Let G be an extremal graph for Kp on n vertices. We want to show
that G is complete multipartite graph (since we then know that is has to be
T p−1 (n)).
Suppose that it is not. Then non-adjacency is not an equivalence relation
on V (G), and therefore, there are vertices v1 , v2 , v3 such that v1 v2 , v2 v3 6∈
V (G) and v1 v3 ∈ E(G). If deg(v1 ) > deg(v2 ) we can delete v2 and duplicate
v1 to get another graph with the same number of vertices but more edges that
still does not contain Kp . This contradict the assumption that G is extremal.
Hence, deg(v1 ) ≤ deg(v2 ) and analogue deg(v3 ) ≤ deg(v2 ). If we delete both
v1 and v3 and then duplicate v2 twice we once again get a Kp -free graph on
the same number of vertices as G but with more edges, which contradicts
the assumption.
A classical theorem in extremal graph theory by Turán is the following.
Theorem B.4 (Turán 1941 [39]).
ex(n, Kp ) ≤ 1 −
2
n
1
p−1 2
It tells us the minimum number of edges a graph must have to guarantee
the existence of p-clique.
Another important question is how tr−1 (n) behaves asymtotically. The
following lemma is not difficult to prove (see [8] for a complete proof).
Lemma B.5.
−1
n
r−2
lim tr−1 (n)
=
n→∞
r−1
2
39
C
Ramsey Theory
The famous pigeonhole principle states that if n + 1 letters are placed in n
pigeonholes, the some pigeonhole must contain more than one letter. A very
famous theorem by Ramsey generalizes this principle and has given rise to a
whole area of combinatorics known as Ramsey theory. The theorem basically
tells us that if we have a large structure and try to partition it, a certain
substructure must occur. Motzkin described this with his famous words:
Total disorder is impossible.
Definition C.1 (Ramsey number). Given a set S and a natural number
r, we say that a subset T ⊂ S is homogeneous under a colouring of Sr if all
r-sets in T have the same colour. If the colour is i, we say that the set is
i-homogeneous.
Let p1 , . . . , pk ∈
N. If there is a smallest number N ∈ N, such that every
[N ]
k-colouring of r results in an i-homogeneous set of size pi for some i ∈ [k],
we call it the Ramsey number R(p1 , . . . , pk ; r).
Theorem C.2 (Ramsey 1930). For every choice of r and p1 , . . . , pk the
number R(p1 , . . . , pk ; r) exists.
For a more comprehensive introduction to Ramsey theory see [17]. A
shorter introduction to the subject is given in the chapters on Ramsey theory
in [6] and [41].
D
NP-completeness
Definition D.1 (NP). A decision problem is in NP if it can be solved by a
non-deterministic Turing Machine in polynomial time. If the complement of
a problem is in NP, we say that the problem is in co-NP.
Definition D.2 (NP-Complete). A decision problem is NP-complete if it
is in NP and all other problems in NP can be reduced to this problem in
deterministic polynomial time.
A reduction r from problem A another problem B is a function such that
if x is a valid instance of A, then r(x) is a valid instance of B and finally A
answers ‘yes’ on x iff B does the same on r(x).
To prove that a decision problem M is NP-complete, it is sufficient to show
that there exists a polynomial time reduction from another NP-complete
problem K to M. This is obvious since we know that all problems in NP can
be reduced to K in polynomial time and hence, we can also reduce them to
40
M in polynomial time by composing the two reductions (as the composition
of polynomial time reductions yields a polynomial time reduction).
There are two ways to prove that a decision problem M 0 is co-NP-complete.
One is to find a polynomial time reduction from another co-NP-complete
problem to M 0 . The other is to find a polynomial time reduction from a
NP-complete problem to the complement of M 0 .
For a more detailed introduction to the subject, see [19] and [42].
41
References
[1] The Combinatorics of Patterns in Subsets and Graphs, lecture notes,
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