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