Introduction to Szemerédi regularity lemma

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Introduction to Szemer´ edi’s Regularity Lemma Abner Chih Yi Huang Oct. 15, 2008 CSBB Lab, CS, NTHU 1/1

Transcript of Introduction to Szemerédi regularity lemma

Introduction to Szemeredi’s Regularity Lemma

Abner Chih Yi Huang

Oct. 15, 2008

CSBB Lab, CS, NTHU

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Szemeredi’s Theorem

Theorem (Szemeredi’s Theorom)

Let k be a positive integer and let 0 < δ < 1.

Then there exists a positive integer

N = N(k , δ), such that for every

A ⊂ 1, 2, ..., N, |A| ≥ δN, A contains an

arithmetic progression of length k.

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For an instance, N(3, 1/2) =?

Consider 1, 2, 3, 4, 5, 6, 7, 81, 4, 5, 8, 2, 3, 6, 7

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For an instance, N(3, 1/2) =?

Consider 1, 2, 3, 4, 5, 6, 7, 81, 4, 5, 8, 2, 3, 6, 7Consider 1, 2, 3, 4, 5, 6, 7, 8, 91, 4, 5, 8, ?

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For an instance, N(3, 1/2) =?

Consider 1, 2, 3, 4, 5, 6, 7, 81, 4, 5, 8, 2, 3, 6, 7Consider 1, 2, 3, 4, 5, 6, 7, 8, 91, 4, 5, 8, ?1, 4, 5, 8, 2, 1, 4, 5, 8, 3, 1, 4, 5, 8, 6, 1, 4, 5, 8, 7,1, 4, 5, 8, 9

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What makes thisinteresting?

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Related to a famous result...

Theorem

The primes contain arbitrarily long

arithmetic progressions.

(Terence Tao and Ben J. Green, 2004)

The arithmetic progressions withlength 23,

56211383760397+44546738095860k ,

is found at 2004, too.

Terence Tao (2006 Fields Medalist)

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Well, it is important.

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Well, it is important.

Three Fields Medalists had worked on this problem, Roth (1958), Gowers(1998), Tao (2006).

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But, this is none of mybusiness.

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But, this is none of mybusiness.

Is anyone here a mathematician or a number theorist?

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Really?

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Really?

TCS: pseudo-random numbers, PCP constructions, communicationcomplexity, sub-linear time algorithms, etc.

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The Origin of Szemeredi’s Theorem

Theorem (Van der Waerden’s theorem, 1927)

For any given positive integers r and k, there is

some number N such that if the integers

1, 2, · · · , N are colored, each with one of r

different colors, then there are at least k integers in

arithmetic progression all of the same color.

It is named VDW.

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The Origin of Szemeredi’s Theorem

Theorem (Van der Waerden’s theorem, 1927)

For any given positive integers r and k, there is some number Wsuch that if the integers 1, 2, · · · ,W are colored, each with oneof r different colors, then there are at least k integers in arithmeticprogression all of the same color.

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The Origin of Szemeredi’s Theorem

Theorem (Van der Waerden’s theorem, 1927)

For any given positive integers r and k, there is some number Wsuch that if the integers 1, 2, · · · ,W are colored, each with oneof r different colors, then there are at least k integers in arithmeticprogression all of the same color.

Fact 1: The constants W = W (k, r) are called van derWaerden numbers.

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The Origin of Szemeredi’s Theorem

Theorem (Van der Waerden’s theorem, 1927)

For any given positive integers r and k, there is some number Wsuch that if the integers 1, 2, · · · ,W are colored, each with oneof r different colors, then there are at least k integers in arithmeticprogression all of the same color.

Fact 1: The constants W = W (k, r) are called van derWaerden numbers.

E.g., N(3, 2) > 8

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The Origin of Szemeredi’s Theorem

Theorem (Van der Waerden’s theorem, 1927)

For any given positive integers r and k, there is some number Wsuch that if the integers 1, 2, · · · ,W are colored, each with oneof r different colors, then there are at least k integers in arithmeticprogression all of the same color.

Fact 1: The constants W = W (k, r) are called van derWaerden numbers.

E.g., N(3, 2) > 81, 2, 3, 4, 5, 6, 7, 8

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The Origin of Szemeredi’s Theorem

Theorem (Van der Waerden’s theorem, 1927)

For any given positive integers r and k, there is some number Wsuch that if the integers 1, 2, · · · ,W are colored, each with oneof r different colors, then there are at least k integers in arithmeticprogression all of the same color.

Fact 1: The constants W = W (k, r) are called van derWaerden numbers.

E.g., N(3, 2) > 81, 2, 3, 4, 5, 6, 7, 81, 2, 3, 4, 5, 6, 7, 8, 9

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The Origin of Szemeredi’s Theorem

Theorem (Van der Waerden’s theorem, 1927)

For any given positive integers r and k, there is some number Wsuch that if the integers 1, 2, · · · ,W are colored, each with oneof r different colors, then there are at least k integers in arithmeticprogression all of the same color.

Fact 1: The constants W = W (k, r) are called van derWaerden numbers.

E.g., N(3, 2) > 81, 2, 3, 4, 5, 6, 7, 81, 2, 3, 4, 5, 6, 7, 8, 9

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VDW Numbers

Fact 2: No formula for N = N(k , r) is known. It

is existence thereom.

W (3, 2) = 9

W (3, 3) = 27

W (3, 4) = 76

W (5, 2) = 178

W (6, 2) = 1132

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VDW Numbers

Fact 2: No formula for N = N(k , r) is known. It

is existence thereom.

W (3, 2) = 9

W (3, 3) = 27

W (3, 4) = 76

W (5, 2) = 178

W (6, 2) = 1132

Theorem

For all k, r , W (k, r) ≥ r (k−1)/2 ·√

k.

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History

van der Waerden

Erdos, Turan, 1936

Szemeredi, 1975

Combinatorics

Furstenberg, 1977

Ergodic theory

Gowers, 2001

Fourier analysis

Tao, Green

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History

Theorem (Roth’s theorem, 1953)

Every set of integers of positive density contained infinitely manyprogressions of length three.

It is the special case of Szemeredi’s Theorom for k = 3. Rothproved this by number theoretic method.

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History

Theorem (Roth’s theorem, 1953)

Every set of integers of positive density contained infinitely manyprogressions of length three.

It is the special case of Szemeredi’s Theorom for k = 3. Rothproved this by number theoretic method.

Theorem (Szemeredi, 1969)

Every set of integers of positive density contained infinitely manyprogressions of length four.

At 1975 Szemeredi proved Szemeredi’s Theorem.

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History

van der Waerden

Erdos, Turan, 1936

Szemeredi, 1975

Combinatorics

Furstenberg, 1977

Ergodic theory

Gowers, 2001

Fourier analysis

Tao, Green

Furstenberg: Wolf prize; Tao/Green, Roth, Gowers: Fields Medal. Szemeredi:

Polya prize

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Why Szemeredi?

Why Szemeredi?

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Does it familiar?

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Does it familiar?

Ramsey number!

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Ramsey number

Definition (Version 1)

The Ramsey number is the minimum number of verticesR = R(m, n) such that all undirected simple graphs oforder v contain a clique of order m or an independent setof order n.

Definition (Version 2)

The Ramsey number R(m, n) is the smallest size ofcomplete graph such that any coloring of edges by twocolors blue and red will exist either a red Km or a blue Kl

where Ki with color means the color of edges of completesubgraph Ki are all the same.

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Ramsey K5

Figure: A 2-colouring of K5 with no monochromatic K3.

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Ramsey Theory

Theorem (Ramsey, 1930)

In any colouring of the edges of a sufficiently large

complete graph, one will find monochromatic

complete subgraphs.

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Ramsey Theory

Theorem (Ramsey, 1930)

In any colouring of the edges of a sufficiently large

complete graph, one will find monochromatic

complete subgraphs.

Unlike Szemeredi’s Theorem, Ramsey’s theorem isproved very early. And not like the Szemeredi’s

Number, there are many result of Ramsey Number.

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”Aliens invade the earth andthreaten to obliterate it in ayear’s time unless humanbeings can find the Ramseynumber for red five and bluefive. We could marshal theworld’s best minds and fastestcomputers, and within a yearwe could probably calculatethe value. If the aliensdemanded the Ramsey numberfor red six and blue six,however, we would have nochoice but to launch apreemptive attack.” —Graham, Ronald L. and JoelH. Spencer. Ramsey Theory.Scientific American July 1990:112-117

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Ramsey Numbers

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Back to Szemeredi’s Theorem

Why Szemeredi?

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Back to Szemeredi’s Theorem

Why Szemeredi?

His combinatorial thecnique connected additivecombinatorics to graph theory. It starts the fashion

to connect additive combinatorics to other fields.

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Today’s Topic

Theorem (Szemeredi’s Regularity Lemma, 1978)

For every ε > 0 and positive integer t, there exists

two integers M(ε, t) and N(ε, t) such thatFor every graph G (V , E ) with at least N(ε, t)vertices, there is a partition (V0, V1, V2, . . . , Vk)of V with:

t ≤ k ≤ M(ε, t),|V0| ≤ εn, and|V1| = |V2| = . . . = |Vk |

such that at least (1− ε)(k2

)of pairs (Vi , Vj) are

ε-regular.

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The reason of talking Szemeredi’s Regularity Lemma

It is the tool of proving Szemeredi’s theorem.

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The reason of talking Szemeredi’s Regularity Lemma

It is the tool of proving Szemeredi’s theorem.

It is a fundamental tool in extremal graphtheory.

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The reason of talking Szemeredi’s Regularity Lemma

It is the tool of proving Szemeredi’s theorem.

It is a fundamental tool in extremal graphtheory.

And the most important reason is:

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The reason of talking Szemeredi’s Regularity Lemma

It is the tool of proving Szemeredi’s theorem.

It is a fundamental tool in extremal graphtheory.

And the most important reason is:

I can’t talk about things

beyond my intelligence!

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Density of bipartite graphs

Definition

Given a bipartite graph G = (A,B ,E ), E ⊂ A × B . The

density of G is defined to be d(A,B) = e(A,B)|A|·|B| , where

e(A,B) is the number of edges between A, B .

A perfect matching of G has density 1/n if |A| = |B | = n.

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Density of bipartite graphs

Definition

Given a bipartite graph G = (A,B ,E ), E ⊂ A × B . The

density of G is defined to be d(A,B) = e(A,B)|A|·|B| , where

e(A,B) is the number of edges between A, B .

A perfect matching of G has density 1/n if |A| = |B | = n.

d(A,B) = 1 If G is complete bipartite.

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Density of bipartite graphs

Definition

Given a bipartite graph G = (A,B ,E ), E ⊂ A × B . The

density of G is defined to be d(A,B) = e(A,B)|A|·|B| , where

e(A,B) is the number of edges between A, B .

A perfect matching of G has density 1/n if |A| = |B | = n.

d(A,B) = 1 If G is complete bipartite.

Extend this idea by cuts.

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ε-regular pair

DefinitionLet ε > 0. Given a graph G and two disjoint vertexsets A ⊂ V , B ⊂ V , we say that the pair (A, B) is

ε-regular if for every X ⊂ A and Y ⊂ B satisfying|X | ≥ ε|A| and |Y | ≥ ε|B |, we have

|d(X , Y ) − d(A, B)| < ε.

If G = (A, B , E ) is a complete bipartite graph,then (A, B) is ε-regular for every ε > 0.

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ε-regular pair (contd.)

1/2-regular 1/2-irregular 1/2-irregular

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Facts about THE Lemma

Fact (Convexity)

Given a graph G and two disjoint vertex sets A, B,

for all integers k ≤ |A|, l ≤ |B |, we have

d(A, B) =1

(|A|k

)(|B|l

)

X⊂A,Y⊂B,|X |=k ,|Y |=l

d(X , Y )

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Facts about THE Lemma

Fact (most degrees into a large set are large)

If (A, B) are ε-regular pair with density d, then for

Y ⊂ B , |Y | > ε|B | we have

#x ∈ A | deg(x , Y ) ≤ (d − ε)|Y | ≤ ε|A|

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Facts about THE Lemma

Fact (most degrees into a large set are large)

If (A, B) are ε-regular pair with density d, then for

Y ⊂ B , |Y | > ε|B | we have

#x ∈ A | deg(x , Y ) ≤ (d − ε)|Y | ≤ ε|A|

Assume that it is larger than ε|A|. Then we canpick these vertices as X , and any Y ⊂ B . Hence

d(X , Y ) ≤ |X |(d−ε)|Y ||X ||Y | = (d − ε). It contradicts,

since d − d(X , Y ) ≥ ε.

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Facts about THE Lemma

Fact (Intersection Lemma)

Let (A,B) are ε-regular pair with density d. IfY ⊂ B , (d − ε)l−1|Y | > ε|B |, l ≥ 1, we have

]x = (x1, · · · , xl ) | xi ∈ A, (Y∩(∩xi∈xN(xi )) ≤ (d−ε)l |Y | ≤ lε|A|l

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Facts about THE Lemma

Fact (Intersection Lemma)

Let (A,B) are ε-regular pair with density d. IfY ⊂ B , (d − ε)l−1|Y | > ε|B |, l ≥ 1, we have

]x = (x1, · · · , xl ) | xi ∈ A, (Y∩(∩xi∈xN(xi )) ≤ (d−ε)l |Y | ≤ lε|A|l

Like last lemma, we want to pick some vertices as X , and anyY ⊂ B .

(Y ∩ (∩xi∈xN(xi )) ≤ (d − ε)ε|B |)

Since(|X |

l

)∼

(ε|A|l

)≤ (ε|A|)l ≤ lε|A|l , we can choose enough

distinct x to composite X . Then, by similar argument, itcontradicts!

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Facts about THE Lemma

Fact (Slicing Lemma)

Assume that

(A, B) is a ε-regular and d(A, B) = d, and

A′ ⊂ A and B ′ ⊂ B satisfy |A′| ≥ γ|A| and

|B ′| ≥ γ|B | for some γ ≥ ε,

then

(A′, B ′) is a ε′-regular where max2ε, γ−1ε, and

d(A′, B ′) ≥ d − ε or d(A′, B ′) ≤ d + ε.

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Slicing Lemma (Cont.)

Consider A′′ ⊂ A′ and B ′′ ⊂ B ′, s.t.|A′′| ≥ ε

γ |A′| ≥ εγ · γ|A| ≥ ε|A| and

|B ′′| ≥ εγ |B ′| ≥ ε

γ · γ|B | ≥ ε|B |.This gives good property, e.g., d(A′′,B ′′) is bounded, i.e.,|d(A,B) − d(A′′,B ′′)| < ε.

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Slicing Lemma (Cont.)

Consider A′′ ⊂ A′ and B ′′ ⊂ B ′, s.t.|A′′| ≥ ε

γ |A′| ≥ εγ · γ|A| ≥ ε|A| and

|B ′′| ≥ εγ |B ′| ≥ ε

γ · γ|B | ≥ ε|B |.This gives good property, e.g., d(A′′,B ′′) is bounded, i.e.,|d(A,B) − d(A′′,B ′′)| < ε.

If |d(A′, B ′) − d(A′′, B ′′)| > 2ε, then it is possible that|d(A, B) − d(A′′, B ′′)| > ε.Nonetheless, if we choose large enough A′′, B ′′, s.t.|A′′| ∼ |A′| ≥ γ|A| ≥ ε|A|, then the ε-regularity might fail!

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Szemeredi’s Regularity Lemma

Theorem (Szemeredi’s Regularity Lemma, 1978)

For every ε > 0 and positive integer t, there exists

two integers M(ε, t) and N(ε, t) such that

For every graph G (V , E ) with at least N(ε, t)vertices, there is a partition (V0, V1, V2, . . . , Vk)of V with:

t ≤ k ≤ M(ε, t),exceptional set |V0| ≤ εn, and|V1| = |V2| = . . . = |Vk |

such that at least (1− ε)(k2

)of pairs (Vi , Vj) are

ε-regular.

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Figure: Partition (V0, V1, V2, . . . , Vk ) of V ; or (V1, V2, . . . , Vk ) and∀i , j , ||Vi | − |Vj || ≤ 1. Note that the partition which consists of singletonsis ε-regular but trivial.

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Proof of Szemeredi’s Regularity Lemma[2, 8]

1 Partition vertices into k = max1ε , t sets with equal size.

2 If this partition satisfy the requirement, this lemma holds.Otherwise, refine ε-irregular sets according to the relationshipamong other sets.

Figure: Refine k sets to at most k · 2k−1 sets.

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Proof of Szemeredi’s Regularity Lemma

3 Divide the sets of new partition into smaller sets of sizen

k(2k−1)2, and recombine those small sets to larger sets of size

nk2k−1 . And the new size of partition is k2k−1 = k ′.

4 Repeats to refine partition of sizek ′2k′−1 = (k2k−1) · 2k2k−1−1 ⇒ . . . until regularity holds.

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Proof of Szemeredi’s Regularity Lemma

3 Divide the sets of new partition into smaller sets of sizen

k(2k−1)2, and recombine those small sets to larger sets of size

nk2k−1 . And the new size of partition is k2k−1 = k ′.

4 Repeats to refine partition of sizek ′2k′−1 = (k2k−1) · 2k2k−1−1 ⇒ . . . until regularity holds.

Claims

1 We only need to refine the partition finite times s = s(ε, t).

2 According to s, we can calculate upper bound of the size ofpartition M(s) = M(s(ε, t)) = M(ε, t), for |V | ≥ M(s).

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Proof of Szemeredi’s Regularity Lemma

Define the potential function

Φ(A,B) =|A||B ||V |2 d2(A,B)

for A,B ⊂ V ,A ∩ B = ∅.Define Φ(P) = 1

2

A,B∈P,A6=B Φ(A,B), where P denotespartition of V . Φ(P) ≤ 1.

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Proof of Szemeredi’s Regularity Lemma

Define the potential function

Φ(A,B) =|A||B ||V |2 d2(A,B)

for A,B ⊂ V ,A ∩ B = ∅.Define Φ(P) = 1

2

A,B∈P,A6=B Φ(A,B), where P denotespartition of V . Φ(P) ≤ 1.

Lemma

If P is not ε-regular, then we can refine P to partition P′ such thatΦ(P′ − Φ(P)) = Ω(ε5), and elements in P are of the same size.

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Proof of Szemeredi’s Regularity Lemma

In the procedure of refinement, we recombine those small setsof size |Ci |

k(2k−1)2to the desired partition of size k2k−1.

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Proof of Szemeredi’s Regularity Lemma

In the procedure of refinement, we recombine those small setsof size |Ci |

k(2k−1)2to the desired partition of size k2k−1.

Denote C ′0 as the new exceptional set. We have

|C ′0| ≤ |C0| +

|Ci |(2k−1)2

× k2k−1

= |C0| +|Ci |2k−1

× k

≤ |C0| +|V |2k−1

It shows that, after each refinement, the size of exceptionalset grows up by |V |

2k−1 .

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Proof of Szemeredi’s Regularity Lemma

Define the function ρ(x) = x4x−1. We have the upper boundof partition size.

M = ρ(s)(k)

where s is the upper bound of iterations, i.e.,

ρ(s)(x) =

s︷ ︸︸ ︷

ρ(ρ(· · · (ρ(·))))

,and k ≥ t is the size of initial partition.

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Proof of Szemeredi’s Regularity Lemma

For the requirement, the size of exceptional set is less thanε|V |, we can approximate the summation of all increments byfollowing formula.

|C0| + s|V |2k−1

≤ ε|V | ⇒ k + s|V |2k−1

≤ ε|V |

where s is the upper bound of iterations, and k ≥ t is the sizeof initial partition.

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Proof of Szemeredi’s Regularity Lemma

Since we pick k as maxε−1, t, we have

k

ε − s2k−1

≤ |V | ∼ k

ε − O(ε−5)

2ε−1−1

→ k

ε

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Proof of Szemeredi’s Regularity Lemma

Since we pick k as maxε−1, t, we have

k

ε − s2k−1

≤ |V | ∼ k

ε − O(ε−5)

2ε−1−1

→ k

ε

Hence M(ε, t) = maxρ(s)(maxε−1, t), kε .

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Proof of Szemeredi’s Regularity Lemma

Now, we would like to show that the ε-regular partition of sizeK exists for t ≤ K ≤ M(ε, t).

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Proof of Szemeredi’s Regularity Lemma

Now, we would like to show that the ε-regular partition of sizeK exists for t ≤ K ≤ M(ε, t).

If |V | ≤ M(ε, t), then let K = |V |, i.e., we have a ε-regularpartition, which consists of singletons,P = C0 = ∅,C1, · · · ,Cn.

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Proof of Szemeredi’s Regularity Lemma

Now, we would like to show that the ε-regular partition of sizeK exists for t ≤ K ≤ M(ε, t).

If |V | ≤ M(ε, t), then let K = |V |, i.e., we have a ε-regularpartition, which consists of singletons,P = C0 = ∅,C1, · · · ,Cn.If |V | > M(ε, t), then iteratively refine the partition untilε-regularity holds.

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Any questions?

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Algorithmic Aspect of THE Lemma

Questions

1 How can we check the regularity efficiently?

2 How can we construct ε-regular partitionsefficiently?

3 How can we apply THE lemma?

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Question 1

1 How can we check the regularity efficiently?

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Question 1

1 How can we check the regularity efficiently?

“Deciding if a given partition of an input graph satisfies theproperty guaranteed by the regularity lemma” isco-NP-complete [1].

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Question 2

1 How can we construct ε-regular partitions efficiently?

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Question 2

1 How can we construct ε-regular partitions efficiently?

2 Gowers proved that a tower of 2’s, e.g., 222222

a tower of 2’sof height 5, is necessary.[4] Best bound now,

C log(1/δ)k−1 ≤ N(k, δ) ≤ 22δ−22

k+9

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Question 2

1 How can we construct ε-regular partitions efficiently?

2 Gowers proved that a tower of 2’s, e.g., 222222

a tower of 2’sof height 5, is necessary.[4] Best bound now,

C log(1/δ)k−1 ≤ N(k, δ) ≤ 22δ−22

k+9

It is constructible in O(n2). [5]

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Question 2

1 How can we construct ε-regular partitions efficiently?

2 Gowers proved that a tower of 2’s, e.g., 222222

a tower of 2’sof height 5, is necessary.[4] Best bound now,

C log(1/δ)k−1 ≤ N(k, δ) ≤ 22δ−22

k+9

It is constructible in O(n2). [5]

It is constructible in parallel with polynomial processors onEREW PRAM model and O(log n) time. [1]

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Question 2

1 How can we construct ε-regular partitions efficiently?

2 Gowers proved that a tower of 2’s, e.g., 222222

a tower of 2’sof height 5, is necessary.[4] Best bound now,

C log(1/δ)k−1 ≤ N(k, δ) ≤ 22δ−22

k+9

It is constructible in O(n2). [5]

It is constructible in parallel with polynomial processors onEREW PRAM model and O(log n) time. [1]

There is an randomized algorithm in expected time O(n). [3]

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Question 3

How can we apply THElemma?

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Applications of THE Lemma

Triangle Removal LemmaProperty Testing on Subgraphs

(Informal Definition of Property Testing) For a fixed propertyP and any object O, determine whether O has property P , orwhether O is far from having property P (i.e., far from anyother object having P ).E.g., inversions of sequence, and cycles in tree-like graphs.

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Triangle Removal Lemma

Lemma (Triangle Removal Lemma [7])

For all 0 < δ < 1, there exists ε = ε(δ), such that

for every n-vertex graph G, at least one of the

following is true:

1. G can be made triangle-free by removing < δn2

edges.

2. G has ≥ εn3 triangles.

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Triangle Removal Lemma

What does it mean?

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Triangle Removal Lemma

What does it mean?

A lot of triangles are easy to be detected.Few triangles will be easy to eliminated if we

remove o(n2) edges.Removing o(n2) edges might eliminate o(n3)

triangles.

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Triangle Removal Lemma

What does it mean?

A lot of triangles are easy to be detected.Few triangles will be easy to eliminated if we

remove o(n2) edges.Removing o(n2) edges might eliminate o(n3)

triangles.

We show this lemma by making use of theRegularity Lemma.

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Proof of the Triangle Removal Lemma

The regularity Lemma

For every ε > 0 and positive integer t, there exists two integers M(ε, t) andN(ε, t) such that

For every graph G(V , E) with at least N(ε, t) vertices, there is a partition

(V0, V1, V2, . . . , Vk) of V with:

t ≤ k ≤ M(ε, t),|V0| ≤ εn, and|V1| = |V2| = . . . = |Vk |

such that at least (1 − ε)k

2

of pairs (Vi , Vj) are ε-regular.

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Proof of the Triangle Removal Lemma

The regularity Lemma

For every ε > 0 and positive integer t, there exists two integers M(ε, t) andN(ε, t) such that

For every graph G(V , E) with at least N(ε, t) vertices, there is a partition

(V0, V1, V2, . . . , Vk) of V with:

t ≤ k ≤ M(ε, t),|V0| ≤ εn, and|V1| = |V2| = . . . = |Vk |

such that at least (1 − ε)k

2

of pairs (Vi , Vj) are ε-regular.

Let ε = δ10 and t = 10

δ .

Star with an arbitrary graph G (n ≥ N(ε, t)).

Find a δ10 -regular partition into k = k( δ

10 , 10δ ) blocks.

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Proof of the Triangle Removal Lemma (contd.)

Using the partition we justobtained, we define a reducedgraph G ′ as follows:

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Proof of the Triangle Removal Lemma (contd.)

I: Remove all edges betweennon-regular pairs (at most δ

10n2

edges).

≤ δ10

(k2

)irregular pairs, and at

most ( nk)2 edges between

each pair.

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Proof of the Triangle Removal Lemma (contd.)

II: Remove all edges inside blocks(at most δ

10n2 edges).

k blocks, and each containsat most

(n/k2

)edges,

t ≤ k(n/k2

)≤ n2

k≤ δ

10n2 edges areremoved.

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Proof of the Triangle Removal Lemma (contd.)

III: Remove all edges between pairsof density < δ

2 (at most δ2n2

edges).

≤ δ2 ( n

k)2 edges between a pair

of density < δ2 , and at most

(k2

)such pairs.

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Proof of the Triangle Removal Lemma (contd.)

Totally at most (δ/10 + δ/10 + δ/2)n2 < δn2 edges areremoved.

Thus if G ′ contains no triangle, the first condition of thelemma is satisfied.

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Proof of the Triangle Removal Lemma (contd.)

Totally at most (δ/10 + δ/10 + δ/2)n2 < δn2 edges areremoved.

Thus if G ′ contains no triangle, the first condition of thelemma is satisfied.

Hence we suppose that G ′ contains a triangle and continue tosee the second condition of the lemma.

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Proof of the Triangle Removal Lemma (contd.)

By some technical reasons, we may assume V0 = ∅ and letm = n/k be the size of the blocks (V1,V2, . . . ,Vk).

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Proof of the Triangle Removal Lemma (contd.)

By some technical reasons, we may assume V0 = ∅ and letm = n/k be the size of the blocks (V1,V2, . . . ,Vk).

A triangle in G ′ must go between three different blocks, sayA, B , and C .

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Proof of the Triangle Removal Lemma (contd.)

By some technical reasons, we may assume V0 = ∅ and letm = n/k be the size of the blocks (V1,V2, . . . ,Vk).

A triangle in G ′ must go between three different blocks, sayA, B , and C .

If there is an edge between A and B ⇒ there must be manyedges (by Step III).

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Proof of the Triangle Removal Lemma (contd.)

Since “most degrees into a largeset are large”

≤ m/4 vertices in A have≤ δ

4m neighbors in B≤ m/4 vertices in A have≤ δ

4m neighbors in C

Hence ≥ m/2 vertices in A haveboth ≥ δ

4m neighbors in B and

≥ δ4m neighbors in C .

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Proof of the Triangle Removal Lemma (contd.)

Consider a such vertex from A.

How many edges go between Sand T?

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Proof of the Triangle Removal Lemma (contd.)

Consider a such vertex from A.

How many edges go between Sand T?

S ≥ δ4m and T ≥ δ

4m

d(B, C ) ≥ δ2 and (B, C ) is

δ10 -regularhence e(B, C ) ≥( δ

2 − δ10)|S ||T | ≥ δ3

64m2

Total # triangles≥ δ3

64m2 · m2 = δ3

128k3 n3.

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Is the Triangle Removal Lemma important? YES!

The Triangle Removal Lemma

For all 0 < δ < 1, there exists ε = ε(δ), such that for every n-vertex graph G ,at least one of the following is true:

1. G can be made triangle-free by removing < δn2 edges.

2. G has ≥ εn3 triangles.

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Is the Triangle Removal Lemma important? YES!

The Triangle Removal Lemma

For all 0 < δ < 1, there exists ε = ε(δ), such that for every n-vertex graph G ,at least one of the following is true:

1. G can be made triangle-free by removing < δn2 edges.

2. G has ≥ εn3 triangles.

The graph property “triangle-free” is “testable”. (sampleO(1

ε ) vertices and check whether the induced subgraphcontains triangles?)

Yet the complexity has dependence of towers of δ.

e.g., 128k3

δ3 , k is tower of 2’s of height depending on O(1/δ).

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Is the Triangle Removal Lemma important? YES!

T. Tao said,

This discovery opened up for the first time thepossibility that Szemeredi type theorems could beproven by purely graph-theoretical techniques,discarding almost entirely the additive structure ofthe problem.

It is easy to prove Roth’s Theorem by Triangle RemovalLemma.

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Erdos and Combinatorical Proof

Erdos did receive the ColePrize of the AmericanMathematical Society in 1951for his many papers on thetheory of numbers, and inparticular for the paper On anew method in elementarynumber theory which leads toan elementary proof of theprime number theorempublished in the Proceedingsof the National Academy ofSciences in 1949.

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Summary

Only for Huge size graphs.

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Summary

Only for Huge size graphs.

Only for dense graphs. (Sparse graph if

|E | = o(|V |2))

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Summary

Only for Huge size graphs.

Only for dense graphs. (Sparse graph if

|E | = o(|V |2))

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Summary

Only for Huge size graphs.

Only for dense graphs. (Sparse graph if

|E | = o(|V |2))Pure theoretical result. Useless in practice.

However, useful in computational complexity.[6]

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Thanks to ...

Thank YOU&

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Thanks to ...

Thanks to Joseph, Chuang-Chieh Lin

This slides is modified from his one.

He is Ph.D. student of professorMaw-Shang Chang,Computation Theory Laboratory,Dept. Computer Science and InformationEngineering,

National Chung Cheng University, Taiwan with his wife.

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References I

Alon, N., Duke, R. A., Lefmann, H., Rodl, V., and

Yuster, R.

The algorithmic aspects of the regularity lemma.Journal of Algorithms 16, 1 (January 1994), 80–109.

Diestel, R.

Graph Theory, vol. 173 of Graduate Texts in Mathematics.Springer-Verlag, Heidelberg, July 2005.

Frieze, A., and Kannan, R.

The regularity lemma and approximation schemes for denseproblems.In FOCS ’96: Proceedings of the 37th Annual Symposium onFoundations of Computer Science (Washington, DC, USA,1996), IEEE Computer Society, p. 12.

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References II

Gowers, W.

Lower bounds of tower type for szemeredi’s uniformity lemma.Geometric And Functional Analysis 7, 2 (May 1997), 322–337.

Kohayakawa, Y., Rodl, V., and Thoma, L.

An optimal algorithm for checking regularity.In SODA ’02: Proceedings of the thirteenth annualACM-SIAM symposium on Discrete algorithms (Philadelphia,PA, USA, 2002), Society for Industrial and AppliedMathematics, pp. 277–286.

Komlos, J., and Simonovits, M.

Szemeredi’s regularity lemma and its applications in graphtheory.Tech. Rep. 96-10, Center for Discrete Mathematics andTheoretical Computer Science (DIMACS), 1996.

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References III

Ruzsa, and Szemeredi.Triple systems with no six points carrying three triangles.In Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely,1976), Vol. II (1978), vol. 2 of Colloquia MathematicaSocietatis Janos Bolyai, North-Holland, Amsterdam-New York,pp. 939–945.

Trevisan, L.

Proof of the regularity lemma.online, 8 2007.lecture note of Additive Combinatorics and Computer Science.

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