Abstract Szemerédi's Regularity Lemma Is a Deep Result in Graph Theory

Abstract

Szemer´edi’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 Regularity Lemma 3 3.1 Deﬁnitions...... 3 3.2 Szemer´edi’s Regularity Lemma ...... 4 3.3 Proofoftheregularitylemma ...... 4 3.4 Bounds ...... 10

4 Historical background 12 4.1 vanderWaerden’stheorem ...... 12 4.2 Szemer´edi’stheorem ...... 12 4.3 Theoriginallemma ...... 14

5 An extremal problem 15 5.1 Theregularitygraph ...... 15 5.2 Erd¨os-Stonetheorem ...... 18

6 3-term arithmetic progressions 21 6.1 Induced matching problem ...... 21 6.2 The (6, 3)problem ...... 22 6.3 3-termarithmeticprogressions ...... 24

7 Algorithmic aspects of the regularity lemma 26 7.1 Testingforregularity ...... 26 7.2 Creatingtheregularitypartitions ...... 27 7.3 Algorithmicapplications ...... 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-universalgraphs ...... 33 9.2 Theblow-uplemma...... 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 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 progressionP 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 Sze- merédi’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 r-partite graph where each partition has s vertices is Ks . We denote f ◦n(x) = f f(x), i.e. the n times composed function. ◦···◦ n [n]= 1,...,n and for a < b we let [a, b]= a, a +1,...,b . { } | {z } { }

2 3 The Regularity Lemma

3.1 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 e(X,Y ) d(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 = C k be a partition of { i}i=0 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) C ε V (G) | 0| ≤ | | (ii) C = C = = C | 1| | 2| ··· | k| (iii) At most εk2 of the pairs (C ,C ), 1 i < j k are not ε-regular. i j ≤ ≤ 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 ε- regular 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. C = C = = C = l and C is the exceptional | 1| | 2| ··· | k| 0 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 = C k with m k M(ε, m). ≥ { i}i=0 ≤ ≤ 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 µ ,...,µ > 0 and γ ,...,γ 0 then: 1 k 1 k ≥ ( γ )2 γ2 i i (1) µi ≤ µi P X Proof. This is a simple consequenceP of the Cauchy-Schwarz inequality, 2 2 2 ( aibi) a b , where ai = √µi and bi = ei/√µi. ≤ i i PNow we defineP P 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 ⊂ A B q(A, B)= | || |d2(A, B) n2 and if = C k is a partition of V (G) we define P { i}i=1 q( )= q(C ,C ) P i j i

4 and finally if we have two partitions and of A and B respectivly, then we let A B q( , )= q(A0, B0) A B A0∈A BX0∈B In order to avoid the difficulties that can arise when = C ,C ,...,C P { 0 1 k} where C0 is the exceptional set, we consider C0 as a set of singletons. In other words we let ˜ = C ,...,C v : v C and define q( )= q( ˜). P { 1 k}∪{{ } ∈ 0} P P The basic idea for the proof is rather simple. If the partition of V (G) is not ε-regular, we just refine this partition to get a new partition P0. This will 0 P give a q( ) that is significantly greater than q( ) and it is easy to show that the valueP of q( ) is bounded. So after a boundedP numbers of refinements we will get a ε-regularP 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 be a partition of A and ⊂ A B be partition of B, then q( , ) q(A, B) A B ≥ Proof. Let = A k and B = B l , then we have A { i}i=1 { i}i=1 q( , ) = q(A , B ) A B i j i,j X 1 e2(A , B ) = i j n2 A B i,j i j X | || | ≥ 2 1 ( e(Ai, Bj)) (1) i,j n2 A B P i,j | i|| j| 1 e2(A, B) = P n2 ( A )( B ) i | i| j | j| = q(A,P B) P

Lemma 3.8. If , 0 are partitions of V (G) and 0 reﬁnes then P P P P q( 0) q( ) P ≥ P .

5 k Proof. Let = Ci i=1 and for each Ci let i be the reﬁned partition so 0 = P, then{ } ∈ P C P Ci S q( ) = q(C ,C ) P i j i 0 and let A, B V (G) be disjoint. If (A, B) is not ε-regular, then there exist partitions ⊂= A , A of A and = B , B A { 1 2} B { 1 2} of B such that A B q( , ) q(A, B)+ ε4 | || | A B ≥ n2

Proof. Since (A, B) is not ε-regular, there must be sets A1 A and B1 B where A >ε A and B >ε B such that ⊂ ⊂ | 1| | | | 1| | | d(A, B) d(A , B ) >ε (2) | − 1 1 | Let A = A A and B = B B . Take = A , A and = B , B as 2 \ 1 2 \ 2 A { 1 2} B { 1 2} partitions of A and B. We will now show that and satisfy the lemma. A B 1 e2(A , B ) q( , ) = i j A B n2 A B i,j i j X | || | 1 e2(A , B ) e2(A , B ) = 1 1 + i j n2 A B A B 1 1 i+j>2 i j | || | X | || | ≥ 1 e2(A , B ) e(A, B) e(A , B )2 (1) 1 1 + − 1 1 n2 A B A B A B | 1|| 1| | || |−| 1|| 1| To shorten the notation we write ζ := d(A , B ) d(A, B) and we get 1 1 − A B | 1|| 1|e(A, B)+ ζ A B = e(A , B ) A B | 1|| 1| 1 1 | || | 6 so

1 A B e(A, B) 2 n2q( , ) | 1|| 1| + ζ A B A B ≥ A B A B | 1|| 1| | 1|| 1| | || | 1 A B A B 2 + | || |−| 1|| 1|e(A, B) ζ A B A B A B A B − | 1|| 1| | || |−| 1|| 1| | || | A B 2ζe(A, B) A B = | 1|| 1|e2(A, B)+ | 1|| 1| + ζ2 A B A 2 B 2 A B | 1|| 1| | | | | | || | A B A B 2ζe(A, B) A B + | || |−| 1|| 1|e2(A, B) | 1|| 1| A 2 B 2 − A B | | | | | || | ζ2 A 2 B 2 + | 1| | 1| A B A B | || |−| 1|| 1| e2(A, B) + ζ2 A A ≥ A B | 1|| 1| | || | ≥ e2(A, B) (2) + ε4 A B A B | || | | || | since A ε A and B ε B . | 1| ≥ | | | 1| ≥ | | The main lemma in the proof of the regularity lemma states that if a partition, , has too many irregular pairs to satisfy the deﬁnition of ε-regularity, then aP subpartitioning of all the irregular pairs give an increase of q( ) by P 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 = A k be a (k,c)-equitable ≤ P { i}i=0 partition of V (G), where A0 εn. If is not ε-regular then there is a (l,c0)-equitable partition |0 =|B ≤ l of VP(G) where k l k4k, P { i}i=0 ≤ ≤ n A0 A + | 0|≤| 0| 2k and ε5 q( 0) q( )+ P ≥ P 2

Proof. Let ij be a partition of Ai such that if the pair (Ai, Aj) is ε-regular we let :=A A and if it not, then we use lemma 3.9 to get a partition Aij { i} of A and of A . This gives new partitions of size 2 and we get Aij i Aji j A A q( , ) q(A , A )+ ε4 | i|| j| Aij Aji ≥ i j n2

7 Take i as the minimal partition of Ai that refines every partition ij, i = j, =A v : v A and let be the partition A 6 A0 {{ } ∈ 0} A = A A { 0} Ai i [ of V (G). This is a refinement of . Since we have = k + 1 and we P k−1 |P| disregard the exceptional set A0, we get i 2 and therefore k k2k. Since is not ε-regular, there must|A | ≤be more than εk2 of the≤ |A| pairs ≤ P (Ai, Aj) that are ε-irregular and hence as many ij that uses lemma 3.9 to refine the irregular pairs. Thus, we get by lemmaA 3.7 and 3.9

q( ) = q( , )+ q( , )+ q( ) A Ai Aj A0 Ai Ai i

1 where we get the last inequality from the fact that A0 εn 4 n which 3 | | ≤ ≤ imply that kc 4 n. Finally, we≥ must show that all parts , i 1 can have the same size Ai ∈A ≥ c0 without the exeptional set growing too large from collecting the remaining vertices. Let B ,...,B be disjoint sets of size d = c , chosen as large 1 l b 4k c as possible, such that each Bi is a subset of some A A0 and A0 = V (G) B . Take 0 = B , which is a partition of V∈A\{(G). Lemma} 3.8 gives \ i P i S S ε5 q( 0) q( ) q( )+ P ≥ A ≥ P 2 Since d = c , we have that the number of sets B that is a subset of some b 4k c i A is less than 4k and therefore k l k4k as was given in the lemma. j ≤ ≤ All that remains to be shown is that the exceptional set B0 is not too l large. The sets (Bi)i=1 contains all but at most d vertices from each set A A (since we chose the collection (B )l as large as possible). We ∈A\{ 0} i 1

8 have

B0 A0 + d | | ≤ | | c|A| A + k2k ≤ | 0| 4k ck = A + | 0| 2k n A + ≤ | 0| 2k 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 ≤ ∈ A A q( ) = | i|| j|d2(A , A ) P n2 i j i 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, V k , with m k M, we simply have to construct the partition in { i}i=1 ≤ ≤ the way already mentioned. Let A V (G) be the minimum set such that 0 ⊂ k V (G) A0 and A1,...,Ak0 be any partition of V (G) A0 such that all| the | sets\ have| equal{ size. Then} A k0. Now we iterate\ the procedure | 0| ≤ 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 . M c22 ε−5 ≤   In [15] Gowers proves that there exists graphs for which the size of the parti-  −1/16 tion must be as large as a tower of 2s of size c0ε , 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 non- trivial 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) n(G) > δ 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 deﬁ- nition of ε-regularity, the partition is allowed to have at most εk2 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 aibj 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 progressi∈ on 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

a + it k−1 = a, a + t, a +2t,...,a +(k 1)t Z { }i=0 { − } ⊂ 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

2This 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 Szemer´edi proved the conjecture for k = 4 [33]. It was a major breakthrough when Szemer´edi proved the conjecture for arbitrary k:

Theorem 4.3 (Szemer´edi 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.

3Ergodic 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 Szemer´edi 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 n , m ,N,M N, such that for every bipartite graph G ≥= 0 0 ∈ (A, B, E) where A = n N and B = m M there exist sets Vi A,i < n and V B,j| ε V and 0 0 ⊂ i ⊂ ij | | 1| i| S >ε2 Vij then | | | | d(T,S) >d(V ,V ) δ i ij − and for each u V ∈ ij N(u) V < (d(V ,V )+ δ) V | ∩ i| i ij | i| 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-graph5[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 V = = V { = l. The edge} V V E(R) if and only if the pair (V ,V ) | 1| ··· | k| i j ∈ i j is ε-regular and d(Vi,Vj) d. We call this graph the regularity graph of G. s s≥ Furthermore, let R = RG(ε,l,d) be the graph where each vertex Vi V (R) s ∈ s is replaced by a set Vi of s independent vertices and for any vertex u Vi s s ∈ and v Vj , uv E(R ) if and only if ViVj E(R). In other words, R is ∈ ∈ s ∈ replaced by copies of Ks,s. R 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.

4This 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. 5This 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 ε > 0 such that if H is a graph with∈ ∆(H) ∈ ∆ ≥ 0 ≤ and R (ε,l,d) is any regularity graph of G with ε ε and G ≤ 0 2s l ≥ d∆ then H Rs (ε,l,d) H G ⊂ 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∈ − | |} | || | − e(X,Y ) d(X,Y )=

V = Y Y = v σ(i) i,0 ⊃···⊃ i,m { i} where Y = Y N(v ). We must make sure that the sets does not get too i,j+1 i,j ∩ j small. By lemma 5.3 we can choose the sets such that Y − Y < (d ) | i,j 1|−| i,j| − if Y − < εl and all but at most εl choices of v will satisfy | i,j 1| j

Y (d ε) Y − (5) | i,j| ≥ − | i,j 1|

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 Y > εl for all | i,j| j < m and that we have at least s ways to choose vi Yi,m−1. We know that Y = V = l and hence for all∈j < m, | i,0| | σ(i)| 1 Y ∆εl (d ε)∆l ∆εl (d ε )∆l ∆ε l d∆l s | i,j| − ≥ − − ≥ − 0 − 0 ≥(5) 2 ≥ This also implies that Y εl and Y ∆εl s. | i,j| ≥ | i,j| − ≥ There are many many other stronger versions of lemma 5.2. One is the Key lemma[24]. Let H G denote the number of labelled copies of H in k → k 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, vs), such that d(us, vs) d. Let H R∈s with h vertices, ∆=∆(H) > 0 and let δ = d ε ≥ ⊂ − and δ∆ ε = 0 2 + ∆ If ε ε and s 1 ε m, then ≤ 0 − ≤ 0 H G > (ε m)h k → k 0 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 Erd¨os-Stone theorem One of the classic results in extremal graph theory is Tur´an’s theorem (see Appendix B), which says that

ex(n, Kr)= tr−1(n)

In 1946, Erd¨os and Stone gave a famous generalization of this theorem. Theorem 5.5 (Erd¨os 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 r ∈ ≥ contains Ks as a subgraph if

2 e(G) t − (n)+ δn ≥ r 1 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 lim = − n→∞ n χ(H) 1 2 − 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 by Turán’s theorem. Then Ks Rs and lemma 5.2 therefore tells us⊂ that r ⊂ G Ks G. r ⊂ 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 ≥ 2 e(G) t − (n)+ δn ≥ r 1 edges. Since G is simple we get δ < 1. r Let d = δ and ∆ = ∆(Ks ). Lemma 5.2 then gives an ε0. Apply the regularity lemma with 0 <ε ε , ≤ 0 δ ε< < 1 (6) 2 18 1 m> δ and 1 γ := 2δ ε2 4ε > 0 (7) − − − m The regularity lemma returns an upper bound on the size of the partition M. Assume that 2Ms n (8) ≥ d∆(1 ε) − Let P = V k be an (k, l)-equitable ε-regular partition of G, where m { i}i=0 ≤ k M. Since the exceptional set V does not need to be empty, we get ≤ 0 n kl (9) ≥ and n V n εn 2s l = −| 0| − k ≥ M ≥(8) d∆ Now lemma 5.2 imply Kr G if K R (ε,l,d). Thus, all we need to s ⊂ r ⊂ G show is that R is dense enough to contain K , i.e. d(V ,V ) d for enough G r i j ≥ regular pairs (Vi,Vj). We know that V 1 e(V ) | 0| (εn)2 0 ≤ 2 ≤ 2 and k e(V ,V ) V kl εnkl 0 i ≤| 0| ≤ i=1 X Furthermore there are at most εk2 irregular pairs, each containing at most l2 edges. By the definition of density we get that there are less then dl2 edges l in the ε-regular pairs with density < d. Each Vi, 1 i k has at most 2 edges. All other edges contribute to the edges in R ≤. Hence≤ G 1 l e(G) (εn)2 + εk2l2 + dl2 + k + e(R )l2 ≤ 2 2 G From Appendix B we know that

2 1 2 r 2 1 n t − (n) n − = 1 (10) r 1 ≤ 2 r 1 − r 1 2 − −

19 For suﬃciently large n this imply

1 e(G) 1 (εn)2 εk2l2 dl2 l k e(R) k2 − 2 − − − 2 ≥ 2 1 k2l2 2 2 1 2 2 1 t − (n)+ δn ε n εnkl 1 k2 r 1 − 2 − 2ε d ≥(6),(9) 2 1 n2 − − − k 2 1 t (n) 1 k2 r−1 +2δ ε2 4ε d ≥(9) 2 1 n2 − − − − m 2 −1 1 2 n 1 = k t − (n) 1 + γ (7) 2 r 1 2 − n 1 r 2 > k2 − lemmaB.5 2 r 1 − t − (k) ≥(10) r 1 By Tur´an’s theorem, we get K R . r ⊂ G 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 ∈

1 n2 e(G) > 1 + β − χ(H) 1 2 − we have εn h H G > k → k M(ε) 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 Sze- merédi’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 2 k least cn /4 edges between the ε-regular pairs. We get P = Vi i=0 as the ε-regular partition. { } Denote the matchings composing E(G) by M1,...,Mn and let c G0 = G M : e(M ) < n2 \ i i 2 Since at most cn2/2 edges are deleted, we have e(G0) cn2/4. If V (M ) V c V /8 for some 1 i k, 1 j≥ n we let M 0 be the | j ∩ i| ≤ | i| ≤ ≤ ≤ ≤ j matching we get by deleting all edges from Mj that are incident to Vi. Let

n 00 0 G = Mi i=1 [ Then cn2 e(G00) ≥ 8 k 0 since at most i=1 c Vi /8 = cn/8 edges are deleted in each Mj. For each 00 | | 0 edge in G let (Vi,Vj) be the ε-regular pair that contains it and M be the P l 6See Appendix A

21 0 0 matching. We let A = Vi V (Ml ) and B = Vj V (Ml ) and obviously we c ∩c ∩ have A 8 Vi and B 8 Vj . | | ≥ | | | | ≥ | | 0 Since A and B are both in the induced matching Ml we have

e(A, B) min( A , B ) ≤ | | | | Assume that A is the smallest, then | | min( A , B ) 1 8 d(A, B) | | | | ≤ A B ≤ B ≤ c V | || | | | | j| The regularity lemma tells us that V can become arbitrary large when n | j| grows (k has an upper bound M(ε, m)) and since the pair (Vi,Vj) is ε-regular, we have d(V ,V ) d(A, B) <ε. This imply that d(V ,V ) 2ε and hence | i j − | | i j | ≤ e(G00) v v V (G00):(V ,V ) is irregular + e(V ,V ) ≤ |{ i j ∈ i j }| i j (Vi,Vj )Xε-regular 2 2 εn +2ε Vi Vj < 3εn ≤ ≤ ≤ | || | 1 Xi

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-Szemer´edi 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)

7All 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 ∈ ∈ ≥ M = e v : e E(H) and v e v { \{ } ∈ ∈ } and if deg(v) 2 we let M be empty. ≤ v 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 2 2 ∈ such that if N > N0 every S N with S δN contains a triple of the form (a, b), (a + d, b), (a, b + d⊂) for some| a,| ≥ b, d N and d =0. { } ∈ 6 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 aiaj E(G) iff (i, j) S. { } { } ∈ Partition∈ the edges in G into equivalence classes a b a b iff i+j = k+l. 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 aibl, aibj, akbl such that aibj akbl guarantees that (a, b), (a + d, b), (a, b + d) S. ∼ { } ⊂ It is rather easy to prove that Ajtai-Szemerédi’s theorem imply Sze- merédi’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. ⊂ | |

1 2 Proof. Deﬁne S = (a, b) : a b A and a, b [N] . Since S 2 A > 1 2 { − ∈ 1 2 ∈ } | | ≥ | | 2 (δn) we can apply Theorem 6.3 with 2 δ . If N is suﬃciently large we know 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 Sze- mer´edi’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 = V k of V (G), then the problem of deciding∈ { i}i=0 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 n2 X = Y = n and e(B) = 2 1, the problem of deciding if B contains a | | | | n n − subgraph isomorphic to K 2 , 2 is NP-complete.

26 Let B be a bipartite graph with vertex classes X,Y of size n and e(B)= n2 1 1. Then the claim is that K n n B iﬀ B is not ε-regular for ε = . To 2 − 2 , 2 ⊂ 2 prove the claim, we ﬁrst assume that B is not ε-regular. Then there must 1 exists a pair (C,D), where C X and D Y and C , D 2 n, that is a witness to the irregulariy. Thus⊂ ⊂ | | | | ≥ 1 1 1 d(C,D) d(X,Y ) = d(C,D) + | − | − 2 n2 ≥ 2

and therefore d(C,D) = 1. This implies that K n n B [X Y ]. The other 2 , 2 ⊂ ∪ 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 n n K 2 , 2 then we have an irregular pair. This reduction can be done in linear time and hence the problem is co- NP-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 0 R R A and a function εA : + + such that either correctly asserts that B is → A0 ε-regular, or else gives a witness for the εA(ε)-irregularity. 2 0 In [23] it is shown that the complexity of is O(n ) when εA(ε) = ε20/54232. A similar result is given in [2] where theA 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 which uses linear-sized expanders can be found in [23]. A This lemma leaves open how behaves when B is ε-regular but not ε0- A 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 ε- regular 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.

8It 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 ver- ∈ k tices. Furthermore, let P = Vi i=0 be an equitable partition of V (G) and { } − assume that V > 42k and 4k > 600γ 5. Given witnesses that more than γk2 | i| 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 γ5 q(P 0) q(P )+ . ≥ 20 0 and with an exceptional class V0 where n V 0 V + | 0 |≤| 0| 4k From the above lemma, it is easy to derive a constructive version of the regularity lemma. 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 = V k where m k { i}i=0 ≤ ≤ 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 P = V ,...,V 1 { 0 b} where V = n/b . This implies that V < b. Let k = b. | i| b c | 0| 1 2. For every pair (Vi,Vj) of Pi, verify if is ε-regular or find a witness of 0 the εA(ε)-irregularity.

ki 3. If there are at most ε 2 pairs that are not veriﬁed as ε-regular, then we have the desired partition and the algorithm halts. 0 0 4. Apply Lemma 7.5 where P = Pi, k = ki and γ = εA(ε) to get P as ki the reﬁned partition with 1 + ki4 classes.

ki 0 5. Let ki+1 := ki4 , Pi+1 := P , 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 and A hence the algorithm runs in time O(n2). Another simple algorithm that creates the regularity partition using sin- gular 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 χ(H) 1 δ(G) > − n χ(H) and then ﬁnds a set of (1 ε) n vertex disjoint copies of H in G. − h 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 2 n > n0 vertices and cn 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 E0 E(G) with E0 εn2 such that the subgraph G00 = G E0 have χ(G⊂00) 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 E0 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 l a graph G, let P0 = Vi i=0 be a partition of V (G). We write (X,Y ) P0 if X Y = and for some{ } i = j (0 i, j l) we have X V ,Y V .≺ ∩ ∅ 6 ≤ ≤ ⊂ i ⊂ j

Definition 8.1. Let 0 < η 1. Then G is (P0, η)-uniform if, for all X,Y V (G) with (X,Y ) P and≤ X , Y ηn(G) there is a constant p (0, ⊂1] ≺ 0 | | | | ≥ ∈ such that e (X,Y ) p X Y ηp X Y | G − | || || ≤ | || | Definition 8.2. Let H G be a spanning subgraph of G. For X,Y V (G), let ⊂ ⊂ eH (X,Y ) e (X,Y ) if eG(X,Y ) > 0 dH,G(X,Y )= G ( 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 | | ≥ | | | | ≥ | | d (A, B) d (X,Y ) ε | H,G − H,G | ≤ k Definition 8.4. A k-equitable partition P = Vi i=0 is (ε,H,G)-regular if k { } at most ε 2 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 non- exceptional class of P .

Theorem 8.5 (Regularity lemma for sparse graphs [22]). Let ε > 0 and k , l N. Then there exist η = η(ε,k , l) > 0, K = K(ε,k , l) k and 0 ∈ 0 0 ≥ 0 30 N = N(ε,k0, l), such that for any (P, η)-uniform graph G of order n N, where P = V l is a partition of V (G), and H G is a spanning subgraph≥ 0 { i}i=0 ⊂ of G, there is an (ε,H,G)-regular k-equitable partition of V (G) reﬁning P0 with k k K. 0 ≤ ≤ Remark. The lemma is applied on subgraphs of (P0, η)-uniform graph. The partition P = V l is introduced to handle l-partite graphs and if G = K 0 { i}i=0 n 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 ε- regularity 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

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 T S where i ⊂ i 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 measure- ment 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) ≤ ≤ V (G) such that for every k-uniform hypergraph G, it is possible to partition r into P = S ,...,S where l < t, so that all but at most ε V (G) k are not { 1 l} | | contained E(S ,...,S ) for some i ,...,i k where 1 i < i k l i1 i k 1 ( ) 1 ( ) (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 Szemer´edi’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 desir- able 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 Recent results using the regularity lemma

9.1 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 R¨odl 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 (γ, δ, σ) if, for all S V (G) with S γn we have that R ⊂ | | ≥ S S (σ δ) | | e(G[S]) (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 (n, m) denote the set of all labelled graphs with n vertices and m edges. ItG is easy to see that for all n N and 0 m n we have ∈ ≤ ≤ 2 n (n, m) = 2 . |G | m Furthermore we let (n, m; γ, δ, σ) (n, m) denote the graphs G (n, m) satisfying property R(γ, δ, σ) and ⊂(n, G m; k) (n, m) denotes the∈k-universal G R U ⊂ G 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)= σ n we have ≥ ≤ b 2 c (n, m(n); k) lim |U | =1 n→∞ (n, m(n)) |G | and (n, m(n); γ, δ, σ) lim |R | =1 n→∞ (n, m(n)) |G |

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 no- tion 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 R¨odl [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 N for which every− graph G with n(G) N 0 ≥ 0 that satisﬁes property (γ, δ, σ) is k-universal. R 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 H0 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 (k,l,β,ε) if its vertex set can be partitioned into a partition P = V ,...,VP such that { 1 k} (i) V = = V = l | 1| ··· | k| (ii) (V ,V ), 1 i < j k are ε-regular i j ≤ ≤ (iii) β 0 and l = l (k, β) such that for ≥ 0 0 0 0 every graph G and integer l l0 with property (k,l,β,ε0) we know that G is k-universal. ≥ P

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 (k,l,β,ε) is bounded both above and below, where P 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 δ = max σ+ 1 { δ 1 , 1 σ + δ . We then have − 2 2 − } 1 1 1 1 0 < δ ( σ + δ)= σ δ σ + δ + δ < 1 (11) 2 − 1 ≤ 2 − 2 − − ≤ ≤ 2 1

1 1 Hence, property (γ, δ, σ) imply property (γ, δ1, 2 ). Let β = 2 δ. Assume 1 R 3 R − that σ = 2 and k β . Apply Lemma 9.7 with β and k to get ε0 and l0. Choose ≥ 1 ε = min ,ε {R(k,k,k;2) 0} 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 N = M(ε, m)l 0 1 ε 0 − and k(1 ε) γ = − M(ε, m)

What remain now is to show that this choices for N0 and γ satisfy the con- clusion of the theorem. 1 Suppose that a graph G with n(G) N0 satisﬁes property (γ, δ, 2 ). Let V t be the ε-regular partition of ≥V (G) with m t M(Rε, m) and { i}i=0 ≤ ≤ V = = V = l that the regularity lemma ensures the existence of. | 1| ··· | t| Furthermore, let RG(ε,l, 0) be the reduced graph of G. Then

t e(R (ε,l, 0)) (1 ε) G ≥ − 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 (V ,V ) with 1 i < j R(k,k,k;2) i j ≤ ≤ is ε-regular. Partition [R(k,k,k;2)] into 3 parts, T , T , T so that the pair (i, j) T 2 1 2 3 ∈ 1 iff d(V ,V ) β , (i, j) T iff β < d(V ,V ) < 1 β and (i, j) T iff i j ≤ 2 ∈ 2 2 i j − 2 ∈ 3 d(V ,V ) 1 β . Then there is a set J [R(k,k,k, 2)] of cardinality k i j ≥ − 2 ⊂ such that RG(ε,l, 0)[J] is monochromatic (from the definition of the Ramsey

35 j number), i.e. 2 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

k βl2 l βk2l2 kl2 1 kl e(G0) + k + < δ ≤ 2 2 2 ≤ 4 2 2 − 2 and since 1 εkn n(G0)= kl − = γn ≥ M(ε, m) we have a contradiction to property (γ, δ, 1 ). Hence α =1. If α = 3 we get P 2 6 k β 1 kl e(G0) 1 l2 > + δ ≥ 2 − 2 2 2 and one again we get a contradiction. Therefore α = 2. Since

(1 ε)n l − l ≥ M(ε, m) ≥ 0

0 β we know that G satisﬁes property (k,l, 2 ,ε0) for all l l0. Lemma 9.7 now implies that G0 is k-universal. P ≥

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 (k,l,β,ε) but with the stronger requirement that all the pairs are (ε, δ)-super-regularP and with the upper bound on the density removed.

36 Deﬁnition 9.9. A graph G has property (k,l,β,ε,δ) if its vertex set can be partitioned into a partition P = V ,...,VQ such that { 1 k} (i) V = = V = l | 1| ··· | k| (ii) All (V ,V ), 1 i < j k are (ε, δ)-super-regular i j ≤ ≤ (iii) d(V ,V ) > β for all 1 i < j k i j ≤ ≤ 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 (k,l,β,ε,δ) with l l0 contains all graphs H with ∆(H) ∆ that admitsQ 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 R¨odl [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 conjec- tures (at least asymptotically) regarding e.g. powers of Hamiltonian cycles such as Seymour’s conjecture and P´osa’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 V is the edge set. It is customary to denote the vertex ⊆ 2 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, ) where V is the set of E V vertices and each e an arbitrary subset of the vertex set. If k , i.e. all edges contain k vertices,∈E we say that H is a k-uniform hypergraph.E ⊆ 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) = c(u). The→ chromatic number χ(G) is∈ the smallest ∈ 6 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 H on n vertices is called extremal for H if 6⊃ 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 Tur´an 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 p−1 edges in T (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 K with n vertices and ex(n, K ) edges is 6⊃ p p 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 v1v2, v2v3 V (G) and v v E(G). If deg(v ) > deg(v ) we can delete v and duplicate6∈ 1 3 ∈ 1 2 2 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(v ) deg(v ) and analogue deg(v ) deg(v ). If we delete both 1 ≤ 2 3 ≤ 2 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 Tur´an is the following.

Theorem B.4 (Tur´an 1941 [39]).

1 n2 ex(n, K ) 1 p ≤ − 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. n −1 r 2 lim tr−1(n) = − n→∞ 2 r 1 −

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 S r, we say that a subset T S is homogeneous under a colouring of r 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 k-colouring of [N]∈ results in an i-homogeneous set of size∈ p for some i [k], r i ∈ we call it the Ramsey number R(p ,...,p ; r). 1 k

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].

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