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

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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 num- ber 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 ap- proximation. 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 Erd¨os and Paul Tur´an 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 in- stance, 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 Szemer´edi’s regularity lemma. Szemer´edi’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 auxil- iary lemma in the proof of another similar conjecture by Erd¨os and Tur´an, 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- mer´edi’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 Erd¨os 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 R¨odl regarding uniform edge distribution and universality. A powerful em- bedding lemma is also mentioned. 1 This thesis should be accessible to anyone with a year or two of math- ematical 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 dis- tributed. 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 uni- formity 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 Szemer´edi’s Regularity Lemma Now we state the main theorem1 in this paper. Theorem 3.4 (Szemer´edi’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.
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