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The Regularity Lemma and Applications Ryan Martin [email protected] Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie Mellon University The Regularity Lemma and Applications – p.1/60

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Page 1: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The Regularity Lemma andApplications

Ryan Martin

[email protected]

Zeev Nehari Visiting Assistant Professor

Mathematics Department

Carnegie Mellon University

The Regularity Lemma and Applications – p.1/60

Page 2: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Arithmetic progressions

An arithmetic sequence is a set of integers of theform

� ���

�� � � � �

where � � and

are positive integers.

The Regularity Lemma and Applications – p.2/60

Page 3: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Arithmetic progressions

An arithmetic progression is a set of integers ofthe form

The Regularity Lemma and Applications – p.2/60

Page 4: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Arithmetic progressions

An arithmetic progression is a set of integers ofthe form

� ��

�� � � � �

� �

where � � and

are positive integers.

The Regularity Lemma and Applications – p.2/60

Page 5: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Arithmetic progressions

An arithmetic progression of length�

is a set ofintegers of the form

� ��

�� � � � �

� �

where � � and

are positive integers.

The Regularity Lemma and Applications – p.2/60

Page 6: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Van der Waerden’s theorem

The van der Waerden theorem has a number ofequivalent forms. One is:

THEOREM [van der Waerden, 1927]Let and be positive integers.

The Regularity Lemma and Applications – p.3/60

Page 7: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Van der Waerden’s theorem

The van der Waerden theorem has a number ofequivalent forms. One is:

THEOREM [van der Waerden, 1927]Let

and � be positive integers.

The Regularity Lemma and Applications – p.3/60

Page 8: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Van der Waerden’s theorem

The van der Waerden theorem has a number ofequivalent forms. One is:

THEOREM [van der Waerden, 1927]Let

and � be positive integers. There exists aconstant �

� �

��

(the van der Waerden number)

The Regularity Lemma and Applications – p.3/60

Page 9: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Van der Waerden’s theorem

The van der Waerden theorem has a number ofequivalent forms. One is:

THEOREM [van der Waerden, 1927]Let

and � be positive integers. There exists aconstant �

� �

��

such that if � � �� �

��

and

���

� � � � � ��� � � �

�,

The Regularity Lemma and Applications – p.3/60

Page 10: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Van der Waerden’s theorem

The van der Waerden theorem has a number ofequivalent forms. One is:

THEOREM [van der Waerden, 1927]Let

and � be positive integers. There exists aconstant �

� �

��

such that if � � �� �

��

and

���

� � � � � ��� � � �

�,then some set � contains an arithmeticprogression of length

.

The Regularity Lemma and Applications – p.3/60

Page 11: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The van der Waerden number

The quantity �� �

��

(the van der Waerdennumber) is well-studied.

The Regularity Lemma and Applications – p.4/60

Page 12: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The van der Waerden number

The quantity �� �

��

(the van der Waerdennumber) is well-studied.

Shelah (1988) proved that the van der Waerdennumbers are primitive recursive.

The Regularity Lemma and Applications – p.4/60

Page 13: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The van der Waerden number

The quantity �� �

��

(the van der Waerdennumber) is well-studied.

Gowers (1998) announced that

��

� � � � �� ��� � � � � � �

The Regularity Lemma and Applications – p.4/60

Page 14: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The van der Waerden number

The quantity �� �

��

(the van der Waerdennumber) is well-studied.

Gowers (1998) announced that

��

� � � � �� ��� � � � � � �

But it has yet to be published.

The Regularity Lemma and Applications – p.4/60

Page 15: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The van der Waerden number

The quantity �� �

��

(the van der Waerdennumber) is well-studied.

Gowers (1998) announced that

��

� � � � �� ��� � � � � � �

An argument using a probabilistic technique (theLovász Local Lemma) gives that

�� �

�� �

��

� �

� �� � �

The Regularity Lemma and Applications – p.4/60

Page 16: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Plain language

Our version of van der Waerden says:

If we color the first � � positive integerswith � colors, we get a monochromaticarithmetic progression of length

, as longas ��� is large enough.

Which color?

The Regularity Lemma and Applications – p.5/60

Page 17: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Plain language

Our version of van der Waerden says:

If we color the first � � positive integerswith � colors, we get a monochromaticarithmetic progression of length

, as longas ��� is large enough.

Which color?

The Regularity Lemma and Applications – p.5/60

Page 18: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The average

If we color a set with � colors, then one of thosecolors will contain

� of the set.

That set is a very likely candidate. Is that theone?

The Regularity Lemma and Applications – p.6/60

Page 19: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The average

If we color a set with � colors, then one of thosecolors will contain

� of the set.

That set is a very likely candidate. Is that theone?

The Regularity Lemma and Applications – p.6/60

Page 20: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Dense subsets

More generally, we can ask the following:

If we have a subset of

��

withpositive density, does it have an arithmeticprogression of length

?

The Regularity Lemma and Applications – p.7/60

Page 21: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Dense subsets

Let �

� �

��

. The family � is said to

have positive upper density if there exists an �

such that

�� � � �.

The Regularity Lemma and Applications – p.7/60

Page 22: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Arithmetic progressions and density

Endre Szemerédi proved that positive upperdensity is sufficient for the existence of a

-termarithmetic progression.

THEOREM [Szemeredi, 1975]For every integer and every , thereexists a threshold such that if, forsome , and , then

must contain an arithmetic progression ofterms.

The Regularity Lemma and Applications – p.8/60

Page 23: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Arithmetic progressions and density

Endre Szemerédi proved that positive upperdensity is sufficient for the existence of a

-termarithmetic progression.

THEOREM [Szemeredi, 1975]For every integer

� � �

and every � � �

, thereexists a threshold � � � ���

� �

��

such that if, forsome � �� , � �

��

and

� � � � �, thenmust contain an arithmetic progression of

terms.

The Regularity Lemma and Applications – p.8/60

Page 24: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The proof of Szemerédi’s theorem

The proof itself is quite long, complicated andingenious.

For example, consider a diagram of the structureof the proof.

The Regularity Lemma and Applications – p.9/60

Page 25: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The proof of Szemerédi’s theorem

F9

L 1’F1 F2 F3 L 1

L 4L 2

L 5

L 3

1110

F12 L 6

F F

mvdW t

F

F4 F6

F75

F

D

T

C

0

The Regularity Lemma and Applications – p.9/60

Page 26: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The Regularity Lemma

Buried in the proof of Szemerédi’s theorem is aprimitive version of the Regularity Lemma.

But the Regularity Lemma is not a statementabout number theory, it’s a statement aboutgraphs.

The Regularity Lemma and Applications – p.10/60

Page 27: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Graph definitions

Let’s define some basic graph terms.

We have a set of

is said to be adjacent to if there is an edgebetween them. The degree of , , is thenumber of edges touching .

The Regularity Lemma and Applications – p.11/60

Page 28: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Graph definitions

Let’s define some basic graph terms.

We have a set of vertices,

is said to be adjacent to if there is an edgebetween them. The degree of , , is thenumber of edges touching .

The Regularity Lemma and Applications – p.11/60

Page 29: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Graph definitions

Let’s define some basic graph terms.

We have a set of vertices, connected by edges

is said to be adjacent to if there is an edgebetween them. The degree of , , is thenumber of edges touching .

The Regularity Lemma and Applications – p.11/60

Page 30: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Graph definitions

Let’s define some basic graph terms.

We have a set of vertices, connected by edges

� is said to be adjacent to � if there is an edgebetween them.

The degree of , , is thenumber of edges touching .

The Regularity Lemma and Applications – p.11/60

Page 31: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Graph definitions

Let’s define some basic graph terms.

We have a set of vertices, connected by edges

� is said to be adjacent to � if there is an edgebetween them. The degree of �,

��� � � � �

, is thenumber of edges touching �.

The Regularity Lemma and Applications – p.11/60

Page 32: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Bipartite

A graph is bipartite if we can partition thevertex set into and so that all edges arebetween and :

We denote such a graph as .

The Regularity Lemma and Applications – p.12/60

Page 33: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Bipartite

A graph is bipartite if we can partition thevertex set into and so that all edges arebetween and :

� � � � � � � � � � � �

� � � � � � � � � � � �

��

��

���

��

��

���

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���

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��

���

��

��

���

��

��

��

��

��

��

��

���

We denote such a graph as ��

� �

.

The Regularity Lemma and Applications – p.12/60

Page 34: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Random bipartite graphs

Let

� �

be a bipartite graph.

� �

�� �

� �

The number of edges is: .Define the density to be:

The Regularity Lemma and Applications – p.13/60

Page 35: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Random bipartite graphs

Let

� �

be a bipartite graph.

� �

�� �

� �

The number of edges is: � �

.

Define the density to be:

The Regularity Lemma and Applications – p.13/60

Page 36: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Random bipartite graphs

Let

� �

be a bipartite graph.

� �

�� �

� �

The number of edges is: � �

.Define the density to be:

� �

The Regularity Lemma and Applications – p.13/60

Page 37: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Random bipartite graphs

Let

� �

be a bipartite graph.

� �

�� �

� �

The number of edges is: � �

.Define the density to be:

� �

� �

��

The Regularity Lemma and Applications – p.13/60

Page 38: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Binomial model

For each pair of vertices: � � ,

� � ,

The Regularity Lemma and Applications – p.14/60

Page 39: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Binomial model

For each pair of vertices: � � ,

� � , let � �

bean edge in the graph with probability

� � � �

independently for each such pair.

The Regularity Lemma and Applications – p.14/60

Page 40: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Binomial model

For each pair of vertices: � � ,

� � , let � �

bean edge in the graph with probability

� � � �

independently for each such pair.

I.e., take an � � and

� � and flip a biasedcoin to see whether or not � �

is an edge.

The Regularity Lemma and Applications – p.14/60

Page 41: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Binomial model

For each pair of vertices: � � ,

� � , let � �

bean edge in the graph with probability

� � � �

independently for each such pair.

The average density is

� � �

� �

� ��

�� �

The Regularity Lemma and Applications – p.14/60

Page 42: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Subgraphs

Choose

and

.What is the probability that

� � �

� � � � �

� �

�� �

Easy to compute because is a binomialrandom variable.

The Regularity Lemma and Applications – p.15/60

Page 43: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Subgraphs

Choose

and

.What is the probability that

� � �

� � � � �

� �

�� �

Easy to compute because� � �

� �

is a binomialrandom variable.

The Regularity Lemma and Applications – p.15/60

Page 44: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Subgraphs

Choose

and

.What is the probability that

� � �

� � � � �

� �

�� �

Easy to compute because� � �

� �

is a binomialrandom variable.

I.e., it is the average of coin flips.

The Regularity Lemma and Applications – p.15/60

Page 45: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Chernoff bound

The Chernoff bound gives the probability that thedensity between these subsets differs from

bymore than �.

This is ... tiny ...

The Regularity Lemma and Applications – p.16/60

Page 46: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Chernoff bound

The Chernoff bound gives the probability that thedensity between these subsets differs from

bymore than �.

��� � � � � �

� �

� �

��

This is ... tiny ...

The Regularity Lemma and Applications – p.16/60

Page 47: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Chernoff bound

The Chernoff bound gives the probability that thedensity between these subsets differs from

bymore than �.

��� � � � � �

� �

� �

�� � � � �

��

�� � � � � � � �

This is ... tiny ...

The Regularity Lemma and Applications – p.16/60

Page 48: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Chernoff bound

In fact, if

� � � � � � and

� � � � � �, then we have:

��� � � � � �

� �

� �

�� � � � �

��

��

�� �

This is ... tiny ...

The Regularity Lemma and Applications – p.16/60

Page 49: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Chernoff bound

In fact, if

� � � � � � and

� � � � � �, then we have:

��� � � � � �

� �

� �

�� � � � �

��

��

�� �

This is ... tiny ...

The Regularity Lemma and Applications – p.16/60

Page 50: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Applies simultaneously to all such sets

In fact, the probability that ALL pairs of sets,each with size � � �, have density in

� �

� �

��

approachesas .

The Regularity Lemma and Applications – p.17/60

Page 51: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Applies simultaneously to all such sets

In fact, the probability that ALL pairs of sets,each with size � � �, have density in

� �

� �

��

approaches

as � .

The Regularity Lemma and Applications – p.17/60

Page 52: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Defining regularity

A pair

is �-regular with density�

if

� �

For all and withand,

we have .

So, -regular pairs mimic random pairs.

The Regularity Lemma and Applications – p.18/60

Page 53: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Defining regularity

A pair

is �-regular with density�

if

� �

� For all

and

with

and,

we have .

So, -regular pairs mimic random pairs.

The Regularity Lemma and Applications – p.18/60

Page 54: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Defining regularity

A pair

is �-regular with density�

if

� �

� For all

and

with

� � � � �� �

and

,we have .

So, -regular pairs mimic random pairs.

The Regularity Lemma and Applications – p.18/60

Page 55: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Defining regularity

A pair

is �-regular with density�

if

� �

� For all

and

with

� � � � �� �

and

� � � � �� �

,

we have .

So, -regular pairs mimic random pairs.

The Regularity Lemma and Applications – p.18/60

Page 56: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Defining regularity

A pair

is �-regular with density�

if

� �

� For all

and

with

� � � � �� �

and

� � � � �� �

,we have

� � � �

� �

� � � �.

So, �-regular pairs mimic random pairs.

The Regularity Lemma and Applications – p.18/60

Page 57: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Random graphs

�-regular pairs are tied inexorably to randompairs.

� � � � � � � � � � � �

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��

���

� �

��

In fact, an -regular pair has a surprising numberof properties that random pairs have.

The Regularity Lemma and Applications – p.19/60

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Random graphs

�-regular pairs are tied inexorably to randompairs.

� � � � � � � � � � � �

� � � � � � � � � � � �

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

��

��

��

���

��

��

��

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

��

��

��

��

��

���

��

� � �

� �

��

In fact, an -regular pair has a surprising numberof properties that random pairs have.

The Regularity Lemma and Applications – p.19/60

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Random graphs

�-regular pairs are tied inexorably to randompairs.

� � � � � � � � � � � �

� � � � � � � � � � � �

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

��

��

��

���

��

��

��

��

��

���

��

��

���

��

��

���

��

��

���

��

��

���

��

��

��

��

��

��

��

���

��

� � �

� �

��

In fact, an �-regular pair has a surprising numberof properties that random pairs have.

The Regularity Lemma and Applications – p.19/60

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The Regularity Lemma

The Regularity Lemma ensures that, for anywith enough vertices,

� �

has a partition with:

� a constant number of pieces,

all but one of equal size,

a dense subgraph has all pairs -regular,density either

zero, orbounded below by a constant.

It’s easier to see in a picture.

The Regularity Lemma and Applications – p.20/60

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The Regularity Lemma

The Regularity Lemma ensures that, for anywith enough vertices,

� �

has a partition with:

� a constant number of pieces,

� all but one of equal size,

a dense subgraph has all pairs -regular,density either

zero, orbounded below by a constant.

It’s easier to see in a picture.

The Regularity Lemma and Applications – p.20/60

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The Regularity Lemma

The Regularity Lemma ensures that, for anywith enough vertices,

� �

has a partition with:

� a constant number of pieces,

� all but one of equal size,

� a dense subgraph�

has all pairs �-regular,density either

zero, orbounded below by a constant.

It’s easier to see in a picture.

The Regularity Lemma and Applications – p.20/60

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The Regularity Lemma

The Regularity Lemma ensures that, for anywith enough vertices,

� �

has a partition with:

� a constant number of pieces,

� all but one of equal size,

� a dense subgraph�

has all pairs �-regular,density either

� zero, or

bounded below by a constant.

It’s easier to see in a picture.

The Regularity Lemma and Applications – p.20/60

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The Regularity Lemma

The Regularity Lemma ensures that, for anywith enough vertices,

� �

has a partition with:

� a constant number of pieces,

� all but one of equal size,

� a dense subgraph�

has all pairs �-regular,density either

� zero, or

� bounded below by a constant.

It’s easier to see in a picture.

The Regularity Lemma and Applications – p.20/60

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The Regularity Lemma

The Regularity Lemma ensures that, for anywith enough vertices,

� �

has a partition with:

� a constant number of pieces,

� all but one of equal size,

� a dense subgraph�

has all pairs �-regular,density either

� zero, or

� bounded below by a constant.

It’s easier to see in a picture.

The Regularity Lemma and Applications – p.20/60

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What this does

Before Regularity:

After Regularity, the graph :

-regular, density

is very applicable.

The Regularity Lemma and Applications – p.21/60

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What this does

After Regularity, the graph :

��

��

��

���

��

� ��

��

� �

��

��

��

� ��

��

� �

� �

� �-regular, density � �

� �-regular, density

� �

� not �-regular

� real edge

is very applicable.

The Regularity Lemma and Applications – p.21/60

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What this does

After Regularity, the graph�

:

��

��

��

���

� ��

��

� �

��

��

��

� �

� �

� �-regular, density � �

is very applicable.

The Regularity Lemma and Applications – p.21/60

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Formal definition

THEOREM [Szemeredi, 1978] (Degree Form)

�� � �

,

��

such that,

The Regularity Lemma and Applications – p.22/60

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Formal definition

THEOREM [Szemeredi, 1978] (Degree Form)

�� � �

,

��

such that,if �

and

� � � �

, then

The Regularity Lemma and Applications – p.22/60

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Formal definition

THEOREM [Szemeredi, 1978] (Degree Form)

�� � �

,

��

such that,if �

and

� � � �

, then

� � � � � � � (a partition) with

The Regularity Lemma and Applications – p.22/60

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Formal definition

THEOREM [Szemeredi, 1978] (Degree Form)

�� � �

,

��

such that,if �

and

� � � �

, then

� � � � � � � (a partition) with

The Regularity Lemma and Applications – p.22/60

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Formal definition

THEOREM [Szemeredi, 1978] (Degree Form)

�� � �

,

��

such that,if �

and

� � � �

, then

� � � � � � � (a partition) with

��

�� �

and

��

� � � � ��

��

The Regularity Lemma and Applications – p.22/60

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Formal definition

THEOREM [Szemeredi, 1978] (Degree Form)

�� � �

,

��

such that,if �

and

� � � �

, then

� � � � � � � (a partition) with

��

�� �

and

��

� � � � ��

��

subgraph

� �

The Regularity Lemma and Applications – p.22/60

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Formal definition

THEOREM [Szemeredi, 1978] (Degree Form)

�� � �

,

��

such that,if �

and

� � � �

, then

� � � � � � � (a partition) with

��

�� �

and

��

� � � � ��

��

subgraph

� �

��� � ��� � � � ��� � �� � �

�� �

�� � �

,

� � �

The Regularity Lemma and Applications – p.22/60

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Formal definition

THEOREM [Szemeredi, 1978] (Degree Form)

�� � �

,

��

such that,if �

and

� � � �

, then

� � � � � � � (a partition) with

��

�� �

and

��

� � � � ��

��

subgraph

� �

��� � ��� � � � ��� � �� � �

�� �

�� � �

,

� � �

� no

edges inside a � ,

� �

The Regularity Lemma and Applications – p.22/60

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Formal definition

THEOREM [Szemeredi, 1978] (Degree Form)

�� � �

,

��

such that,if �

and

� � � �

, then

� � � � � � � (a partition) with

��

�� �

and

��

� � � � ��

��

subgraph

� �

��� � ��� � � � ��� � �� � �

�� �

�� � �

,

� � �

� no

edges inside a � ,

� �

� all pairs� �

� ��

, (

) are �-regular,each with a density either

or

� �

.The Regularity Lemma and Applications – p.22/60

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Geometry applications

One application is to look at distances in theplane.Let us be given � points in the plane.

There are

pairs of points.

The Regularity Lemma and Applications – p.23/60

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Geometry applications

One application is to look at distances in theplane.Let us be given � points in the plane.

There are

��

��

� � �

��

pairs of points.

The Regularity Lemma and Applications – p.23/60

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Getting large distances

Let � be a large number divisible by�

.Consider the following subset of

�:

Consider two points, and andcompute their distance:

as long as .Even if , the distance is at least 1.

The Regularity Lemma and Applications – p.24/60

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Getting large distances

Let � be a large number divisible by�

.Consider the following subset of

�:

� � ��

� � � � � � �

� � �

�� � �

Consider two points, and andcompute their distance:

as long as .Even if , the distance is at least 1.

The Regularity Lemma and Applications – p.24/60

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Getting large distances

� � ��

� � � � � � �

� � �

�� � �

Consider two points,

� � ��

� �and

� � �

��

� � �

andcompute their distance:

� �

�� � � �

�� � �

�� � � �

as long as .Even if , the distance is at least 1.

The Regularity Lemma and Applications – p.24/60

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Getting large distances

� � ��

� � � � � � �

� � �

�� � �

Consider two points,

� � ��

� �and

� � �

��

� � �

andcompute their distance:

� �

�� � �

�� �

� �

�� � � �

�� � �

�� � � � � �

�� � �

��

as long as

� � � �.

Even if , the distance is at least 1.

The Regularity Lemma and Applications – p.24/60

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Getting large distances

� � ��

� � � � � � �

� � �

�� � �

Consider two points,

� � ��

� �and

� � �

��

� � �

andcompute their distance:

� �

�� � �

�� �

� �

�� � � �

�� � �

�� � � � � �

�� � �

��

as long as

� � � �.

Even if� � � �

, the distance is at least 1.

The Regularity Lemma and Applications – p.24/60

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A picture of this family�

��

�� �

� � � � � � � � � � �

��

��

��

��

The Regularity Lemma and Applications – p.25/60

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A picture of this family�

��

�� �

� � � � � � � � � � �

��

��

��

��

��

��

��

��

The Regularity Lemma and Applications – p.25/60

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Distances are restricted

Thus, has at least

��

��

pairs with distances in

��

�� �

The Regularity Lemma and Applications – p.26/60

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Best possible, asymptotically

THEOREM [Erdos-Makai-Pach-Spencer, 1991]Given � � �

, there is a � and a positive integer � �

satisfying the following condition:

For any set ( ) withminimal distance at least and for any real ,the number of pairs whose distance is

is at most

The Regularity Lemma and Applications – p.27/60

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Best possible, asymptotically

THEOREM [Erdos-Makai-Pach-Spencer, 1991]Given � � �

, there is a � and a positive integer � �

satisfying the following condition:For any set

�� �

� �

��

� �( � �� ) with

minimal distance at least

and for any real

,the number of pairs

��

� �

�whose distance is

��

� � � �� � � �

� � �

is at most

The Regularity Lemma and Applications – p.27/60

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Best possible, asymptotically

THEOREM [Erdos-Makai-Pach-Spencer, 1991]Given � � �

, there is a � and a positive integer � �

satisfying the following condition:For any set

�� �

� �

��

� �( � �� ) with

minimal distance at least

and for any real

,the number of pairs

��

� �

�whose distance is

��

� � � �� � � �

� � �

is at most �� � �

The Regularity Lemma and Applications – p.27/60

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Best possible, asymptotically

THEOREM [Erdos-Makai-Pach-Spencer, 1991]Given � � �

, there is a � and a positive integer � �

satisfying the following condition:For any set

�� �

� �

��

� �( � �� ) with

minimal distance at least

and for any real

,the number of pairs

��

� �

�whose distance is

��

� � � �� � � �

� � �

is at most �� � �

There is a generalization.

The Regularity Lemma and Applications – p.27/60

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Best possible, asymptotically

THEOREM [Erdos-Makai-Pach, 1993]Given

� � �

and � � �

, there is a � and apositive integer � � satisfying the followingcondition:For any set

�� �

� �

��

� �

( � ��� ) withminimal distance at least

and for any reals

��

��

� � , the number of pairs

��

� �

whosedistance is

��

� � � �� �

���� �

� � �� � � �

is at most�

� � � � �

�� �

�� � � �

The Regularity Lemma and Applications – p.27/60

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Graph theory applications

Most applications are in graph theory.

Define a complete graph on vertices, , to bethe graph with vertices where there is an edgebetween each pair of vertices.

The Regularity Lemma and Applications – p.28/60

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Graph theory applications

Most applications are, of course, in graph theory.

Define a complete graph on vertices, , to bethe graph with vertices where there is an edgebetween each pair of vertices.

The Regularity Lemma and Applications – p.28/60

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Graph theory applications

Most applications are, of course, in graph theory.

Define a complete graph on � vertices, �, to bethe graph with � vertices where there is an edgebetween each pair of vertices.

The Regularity Lemma and Applications – p.28/60

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Graph theory applications

Define a complete graph on � vertices, �, to bethe graph with � vertices where there is an edgebetween each pair of vertices.

� �

� �

� �

� �

��

��

��

��

��

� �

� � � � �

�����

��

��

��

��

� � � � � � � ���������

��

��

� � � � �

������

The Regularity Lemma and Applications – p.28/60

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Theorem of Turán

Turán’s theorem is a classical result. Here is aspecial case.

THEOREM [Turan, 1941]

Let be the complete graph on vertices.

Let , on vertices, contain no as asubgraph.

The Regularity Lemma and Applications – p.29/60

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Theorem of Turán

Turán’s theorem is a classical result. Here is aspecial case.

THEOREM [Turan, 1941]

� Let � be the complete graph on

vertices.

� Let �, on � vertices, contain no � as asubgraph.

The Regularity Lemma and Applications – p.29/60

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Theorem of Turán

Turán’s theorem is a classical result. Here is aspecial case.

THEOREM [Turan, 1941]

� Let � be the complete graph on

vertices.

� Let �, on � vertices, contain no � as asubgraph. Then,

� �

�� �

��

The Regularity Lemma and Applications – p.29/60

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Consequence of Regularity Lemma

THEOREMLet � �

be given and write � �� � � � �

.

There is an such that if is large enoughand a graph has

,

then contains at least copies of .

The Regularity Lemma and Applications – p.30/60

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Consequence of Regularity Lemma

THEOREMLet � �

be given and write � �� � � � �

.There is an

��

such that if � is large enoughand a graph � has

� �

�� � �

��

� ,

then contains at least copies of .

The Regularity Lemma and Applications – p.30/60

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Consequence of Regularity Lemma

THEOREMLet � �

be given and write � �� � � � �

.There is an

��

such that if � is large enoughand a graph � has

� �

�� � �

��

� ,

then � contains at least

�� � � � �

copies of �.

The Regularity Lemma and Applications – p.30/60

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Zero to many

There are graphs � such that

� �

�� �

�� �

�� �

�� � �

��

� �

but � contains no �.

So, just a few edges takes us from zero copies ofto a large number:

Note that is a constant.

The Regularity Lemma and Applications – p.31/60

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Zero to many

There are graphs � such that

� �

�� �

�� �

�� �

�� � �

��

� �

but � contains no �.

So, just a few edges takes us from zero copies of

� to a large number:� �

��

Note that is a constant.

The Regularity Lemma and Applications – p.31/60

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Zero to many

There are graphs � such that

� �

�� �

�� ��

��

� � ��

�� �

but � contains no �.

Just a few edges takes us from zero copies of �

to a large number:� �

��

Note that is a constant.

The Regularity Lemma and Applications – p.31/60

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Zero to many

There are graphs � such that

� �

�� �

�� ��

��

� � ��

�� �

but � contains no �.

Just a few edges takes us from zero copies of �

to a large number:� �

��

Note that�

��

is a constant.

The Regularity Lemma and Applications – p.31/60

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Size of �

Question: How big is this�

��

?

Answer: Huge.

The Regularity Lemma and Applications – p.32/60

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Size of �

Question: How big is this�

��

?

Answer:

Huge.

The Regularity Lemma and Applications – p.32/60

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Size of �

Question: How big is this�

��

?

Answer: Huge.

The Regularity Lemma and Applications – p.32/60

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OK, smart guy, how huge?

Applying the Regularity Lemma to a graphrequires it to have many vertices.

The proof of the Regularity Lemma gives a towerfunction for .

What the proof gives is:

The Regularity Lemma and Applications – p.33/60

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OK, smart guy, how huge?

Applying the Regularity Lemma to a graphrequires it to have many vertices.

The proof of the Regularity Lemma gives a towerfunction for

��

.

What the proof gives is:

The Regularity Lemma and Applications – p.33/60

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OK, smart guy, how huge?

Applying the Regularity Lemma to a graphrequires it to have many vertices.

The proof of the Regularity Lemma gives a towerfunction for

��

.

What the proof gives is:

� � � �� �

� ��� ��� �

��

The Regularity Lemma and Applications – p.33/60

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OK, smart guy, how huge?

Applying the Regularity Lemma to a graphrequires it to have many vertices.

The proof of the Regularity Lemma gives a towerfunction for

��

.

What the proof gives is:

� � � �� �

� ��� ��� �

��

� �� � �

� ��

��� �� �

The Regularity Lemma and Applications – p.33/60

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OK, smart guy, how huge?

Applying the Regularity Lemma to a graphrequires it to have many vertices.

The proof of the Regularity Lemma gives a towerfunction for

��

.

What the proof gives is:

��

��

���

� � � �� �

� ��� ��� �

��

� �� � �

� ��

��� ��� � �

����

��

The Regularity Lemma and Applications – p.33/60

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Does it need to be so big?

Some kind of tower function is necessary.

THEOREM [Gowers, 1997] For any � � �

, there is agraph so that any application of the RegularityLemma requires that the number of clusters is atleast a number which is a tower of twos of heightproportional to

�� � � �

��.

The Regularity Lemma and Applications – p.34/60

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Results are still satisfying

Despite the size requirement, there are still prettyresults.

� A diameter 2 graph is one that has a path oflength 2 between any pair of nonadjacentvertices.

A minimal diameter 2 graph is a diameter 2graph but is not, .

Any complete bipartite graph is minimaldiameter 2.

The Regularity Lemma and Applications – p.35/60

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Results are still satisfying

Despite the size requirement, there are still prettyresults.

� A diameter 2 graph is one that has a path oflength 2 between any pair of nonadjacentvertices.

� A minimal diameter 2 graph is a diameter 2graph but

� � is not,

� � � � �

.

Any complete bipartite graph is minimaldiameter 2.

The Regularity Lemma and Applications – p.35/60

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Results are still satisfying

Despite the size requirement, there are still prettyresults.

� A diameter 2 graph is one that has a path oflength 2 between any pair of nonadjacentvertices.

� A minimal diameter 2 graph is a diameter 2graph but

� � is not,

� � � � �

.

� Any complete bipartite graph is minimaldiameter 2.

The Regularity Lemma and Applications – p.35/60

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Complete bipartite graphs

Consider a complete bipartite graph ���

�:�

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

The Regularity Lemma and Applications – p.36/60

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Complete bipartite graphs

Delete an edge � �.�

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

The Regularity Lemma and Applications – p.36/60

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Complete bipartite graphs

Distance between � and � is�

.�

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

��

��

��

��

���

The Regularity Lemma and Applications – p.36/60

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Minimal diameter 2 theorem

THEOREM [Furedi, 1992]There is a � � such that if � � � and � is aminimal graph of diameter 2, then

� �

�� �

��

Furthermore, equality occurs if and only if

The Regularity Lemma and Applications – p.37/60

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Minimal diameter 2 theorem

THEOREM [Furedi, 1992]There is a � � such that if � � � and � is aminimal graph of diameter 2, then

� �

�� �

��

Furthermore, equality occurs if and only if� �

�� � ��

�� � �

The Regularity Lemma and Applications – p.37/60

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Loebl’s Conjecture

CONJECTURE [Loebl]If � is a graph on � vertices, and at least �

� �

vertices have degrees at least �� �

, thencontains, as subgraphs, all trees with at most �

� �

edges.

The Regularity Lemma and Applications – p.38/60

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Loebl’s Conjecture

CONJECTURE [Loebl]If � is a graph on � vertices, and at least �

� �

vertices have degrees at least �� �

, thencontains, as subgraphs, all trees with at most �

� �

edges.

THEOREM [Ajtai-Komlos-Szemeredi, 1994]For every � � �

, there is a � � such that if � hasat least

��

�� �

vertices of degree at least

��

�� �

, then � contains, as subgraphs, alltrees with at most �

� �

edges.

The Regularity Lemma and Applications – p.38/60

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Loebl’s Conjecture

CONJECTURE [Loebl]If � is a graph on � vertices, and at least �

� �

vertices have degrees at least �� �

, thencontains, as subgraphs, all trees with at most �

� �

edges.

THEOREM [Zhao, 2003?]There is a constant � � so that Loebl’s Conjectureholds for � � � .

The Regularity Lemma and Applications – p.38/60

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Hajnal-Szemerédi

There is a classical theorem, not proven byregularity, that gives a condition for which agraph can be covered by copies of �.

Define

� � �

� � ��� � ��� � � � � � � � � � �

.

THEOREM [Hajnal-Szemeredi, 1969]If , then containsvertex-disjoint copies of .

Let’s just deal with .

The Regularity Lemma and Applications – p.39/60

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Hajnal-Szemerédi

There is a classical theorem, not proven byregularity, that gives a condition for which agraph can be covered by copies of �.

Define

� � �

� � ��� � ��� � � � � � � � � � �

.

THEOREM [Hajnal-Szemeredi, 1969]If

� �

�� �

� �

��

� , then � contains

��

��

vertex-disjoint copies of �.

Let’s just deal with .

The Regularity Lemma and Applications – p.39/60

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Hajnal-Szemerédi

There is a classical theorem, not proven byregularity, that gives a condition for which agraph can be covered by copies of �.

Define

� � �

� � ��� � ��� � � � � � � � � � �

.

THEOREM [Hajnal-Szemeredi, 1969]If

� �

�� �

� �

��

� , then � contains

��

��

vertex-disjoint copies of �.

Let’s just deal with � � �

.

The Regularity Lemma and Applications – p.39/60

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Hajnal-Szemerédi

There is a classical theorem, not proven byregularity, that gives a condition for which agraph can be covered by copies of �.

Define

� � �

� � ��� � ��� � � � � � � � � � �

.

THEOREM [Corradi-Hajnal, 1963]If

� �

�� � � � � �

� , then � contains

�� � �

vertex-disjoint copies of �.

Let’s just deal with � � �

.

The Regularity Lemma and Applications – p.39/60

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A small example

Here the minimum degree is

� �� � � � �

� �

.

� � � � �

��

��

��

��

��

� � � � �

�����

�����

��

��

��

��

��

The Regularity Lemma and Applications – p.40/60

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A small example

Hajnal-Szemerédi says that it can be covered bytriangles ( � ’s).

� � � � �

��

��

��

��

��

� � � � �

�����

�����

��

��

��

��

��

The Regularity Lemma and Applications – p.40/60

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Tripartite version

The following conjecture is a natural extension ofCorrádi-Hajnal, but not a consequence:

PROBLEMLet be a graph that is

-partite (tripartite),

has vertices in each partition,

has each vertex adjacent to vertices ineach of the other classes.

Then contains vertex-disjoint copies of .

In fact, “ ” can be replaced by “ ”, but there isone exceptional case.

The Regularity Lemma and Applications – p.41/60

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Tripartite version

The following conjecture is a natural extension ofCorrádi-Hajnal, but not a consequence:

PROBLEMLet be a graph that is

-partite (tripartite),

� has vertices in each partition,

has each vertex adjacent to vertices ineach of the other classes.

Then contains vertex-disjoint copies of .

In fact, “ ” can be replaced by “ ”, but there isone exceptional case.

The Regularity Lemma and Applications – p.41/60

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Tripartite version

The following conjecture is a natural extension ofCorrádi-Hajnal, but not a consequence:

PROBLEMLet be a graph that is

-partite (tripartite),

� has vertices in each partition,

� has each vertex adjacent to �

�� vertices in

each of the other classes.

Then contains vertex-disjoint copies of .

In fact, “ ” can be replaced by “ ”, but there isone exceptional case.

The Regularity Lemma and Applications – p.41/60

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Tripartite version

The following conjecture is a natural extension ofCorrádi-Hajnal, but not a consequence:

PROBLEMLet be a graph that is

-partite (tripartite),

� has vertices in each partition,

� has each vertex adjacent to �

�� vertices in

each of the other classes.

Then contains vertex-disjoint copies of �.

In fact, “ ” can be replaced by “ ”, but there isone exceptional case.

The Regularity Lemma and Applications – p.41/60

Page 137: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Tripartite version

PROBLEMLet be a graph that is

-partite (tripartite),

� has vertices in each partition,

� has each vertex adjacent to �

�� vertices in

each of the other classes.

Then contains vertex-disjoint copies of �.

In fact, “ � ” can be replaced by “ ”, but there isone exceptional case.

The Regularity Lemma and Applications – p.41/60

Page 138: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

An example

Each vertex is adjacent to at least� � � � � � �

� �

vertices in each of the other classes.

� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

���������������

��������������������

���������������� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

���������������

���������������

��������������������

��

��

��

��

��

��

��

��������������������

��

��

��

��

��

��

��

�� � � � � � � � � � � � � � �

� � � � � � � � � � � � � � �

����������

����������

����������� � � � � � � � � � � � � � �

� � � � � � � � � � � � � � �

����������

����������

����������� � � � � � � � � � � � � � � � � � � �

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

The Regularity Lemma and Applications – p.42/60

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An example

Each vertex is adjacent to at least� � � � � � �

� �

vertices in each of the other classes.

� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

���������������

��������������������

���������������� � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

���������������

���������������

��������������������

��

��

��

��

��

��

��

��������������������

��

��

��

��

��

��

��

�� � � � � � � � � � � � � � �����������

� � � � � � � � � � � � � � �����������

����������� � � � � � � � � � � � � � �

����������� � � � � � � � � � � � � � �

��������������������� � � � � � � � � � � � � � � � � � � �

The Regularity Lemma and Applications – p.42/60

Page 140: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

An example

Each vertex is adjacent to at least� � � � � � �

� �

vertices in each of the other classes.

� � � � � � � � � � � � � � �����������

� � � � � � � � � � � � � � �����������

����������� � � � � � � � � � � � � � �

����������� � � � � � � � � � � � � � �

��������������������� � � � � � � � � � � � � � � � � � � �

The Regularity Lemma and Applications – p.42/60

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The result is tight

Consider the following example. Each vertex isadjacent to 2 in each other piece.

��

��

��

��

��

���

��

��

��

��

��

��

��

���

��

��

���

��

��

���

��

��

���

The Regularity Lemma and Applications – p.43/60

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The result is tight

Without loss of generality, � must be in the redtriangle.

���

��

��

��

��

���

��

��

��

��

��

��

��

���

��

��

���

��

��

���

��

��

��

��

��

��

��

��

���

The Regularity Lemma and Applications – p.43/60

Page 143: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The result is tight

But then, � cannot be in any triangle.

�� ��

��

��

��

��

���

��

��

��

��

��

��

��

���

��

��

���

��

��

���

��

��

��

��

��

��

��

��

���

The Regularity Lemma and Applications – p.43/60

Page 144: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Example generalizes

This example generalizes.For , an odd multiple of 3, there is a graph with

� vertices in each class,

each vertex adjacent to exactlyvertices in each of the two other parts, but

NO subgraph of vertex-disjoint copies of.

There is only one such graph (up toisomorphism) and we call it .

The Regularity Lemma and Applications – p.44/60

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Example generalizes

This example generalizes.For , an odd multiple of 3, there is a graph with

� vertices in each class,

� each vertex adjacent to exactly

� � � � �

vertices in each of the two other parts, but

NO subgraph of vertex-disjoint copies of.

There is only one such graph (up toisomorphism) and we call it .

The Regularity Lemma and Applications – p.44/60

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Example generalizes

This example generalizes.For , an odd multiple of 3, there is a graph with

� vertices in each class,

� each vertex adjacent to exactly

� � � � �

vertices in each of the two other parts, but

� NO subgraph of vertex-disjoint copies of �.

There is only one such graph (up toisomorphism) and we call it .

The Regularity Lemma and Applications – p.44/60

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Example generalizes

This example generalizes.For , an odd multiple of 3, there is a graph with

� vertices in each class,

� each vertex adjacent to exactly

� � � � �

vertices in each of the two other parts, but

� NO subgraph of vertex-disjoint copies of �.

There is only one such graph (up toisomorphism) and we call it

��

� � � �

.

The Regularity Lemma and Applications – p.44/60

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Tripartite

THEOREM [Magyar-M., 2002]Let be a tripartite graph such that

� there are vertices in each partition

each vertex is adjacent to at leastvertices in each of the other parts, and

for some absolute constant .

Then either has a subgraph which isvertex-disjoint triangles, or foran odd integer.

The Regularity Lemma and Applications – p.45/60

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Tripartite

THEOREM [Magyar-M., 2002]Let be a tripartite graph such that

� there are vertices in each partition

� each vertex is adjacent to at least

� � � � �

vertices in each of the other parts, and

for some absolute constant .

Then either has a subgraph which isvertex-disjoint triangles, or foran odd integer.

The Regularity Lemma and Applications – p.45/60

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Tripartite

THEOREM [Magyar-M., 2002]Let be a tripartite graph such that

� there are vertices in each partition

� each vertex is adjacent to at least

� � � � �

vertices in each of the other parts, and

� � for some absolute constant � .

Then either has a subgraph which isvertex-disjoint triangles, or foran odd integer.

The Regularity Lemma and Applications – p.45/60

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Tripartite

THEOREM [Magyar-M., 2002]Let be a tripartite graph such that

� there are vertices in each partition

� each vertex is adjacent to at least

� � � � �

vertices in each of the other parts, and

� � for some absolute constant � .

Then either has a subgraph which isvertex-disjoint triangles, or

foran odd integer.

The Regularity Lemma and Applications – p.45/60

Page 152: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Tripartite

THEOREM [Magyar-M., 2002]Let be a tripartite graph such that

� there are vertices in each partition

� each vertex is adjacent to at least

� � � � �

vertices in each of the other parts, and

� � for some absolute constant � .

Then either has a subgraph which isvertex-disjoint triangles, or � �

� � � �

for

� �

an odd integer.

The Regularity Lemma and Applications – p.45/60

Page 153: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

What’s with �?

Yep, that � is the

��

from the RegularityLemma.

The Regularity Lemma and Applications – p.46/60

Page 154: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Quadripartite?

A partial result in the quadripartite case:

THEOREM [Fischer, 1999]Let be a quadripartite graph with

� vertices in each part,

� each vertex adjacent to at least

� � � � �

vertices in each of the other three parts,

Then there is an absolute constant such thathas a subgraph which is a family of �

vertex-disjoint �’s.

The Regularity Lemma and Applications – p.47/60

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Diagram of Fischer’s result

� � � �

� � � �

� � � �

� � � �

� � � � � � � �

� � � � � � � �

� � � � � � � �

� � � � � � � �

��

���

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��

���

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��

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��

��

���

�����

�����

�����

�����

�����

�����

�����

�����

The Regularity Lemma and Applications – p.48/60

Page 156: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Quadripartite!

THEOREM [M.-Szemeredi, 200?]Let be a quadripartite graph such that

� there are vertices in each partition

each vertex is adjacent to at leastvertices in each of the other parts, and

for some absolute constant .

Then has a subgraph which isvertex-disjoint ’s.

This case has no exceptional graph.

The Regularity Lemma and Applications – p.49/60

Page 157: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Quadripartite!

THEOREM [M.-Szemeredi, 200?]Let be a quadripartite graph such that

� there are vertices in each partition

� each vertex is adjacent to at least

� � � � �

vertices in each of the other parts, and

for some absolute constant .

Then has a subgraph which isvertex-disjoint ’s.

This case has no exceptional graph.

The Regularity Lemma and Applications – p.49/60

Page 158: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Quadripartite!

THEOREM [M.-Szemeredi, 200?]Let be a quadripartite graph such that

� there are vertices in each partition

� each vertex is adjacent to at least

� � � � �

vertices in each of the other parts, and

� � for some absolute constant � .

Then has a subgraph which isvertex-disjoint ’s.

This case has no exceptional graph.

The Regularity Lemma and Applications – p.49/60

Page 159: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Quadripartite!

THEOREM [M.-Szemeredi, 200?]Let be a quadripartite graph such that

� there are vertices in each partition

� each vertex is adjacent to at least

� � � � �

vertices in each of the other parts, and

� � for some absolute constant � .

Then has a subgraph which isvertex-disjoint �’s.

This case has no exceptional graph.

The Regularity Lemma and Applications – p.49/60

Page 160: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Quadripartite!

THEOREM [M.-Szemeredi, 200?]Let be a quadripartite graph such that

� there are vertices in each partition

� each vertex is adjacent to at least

� � � � �

vertices in each of the other parts, and

� � for some absolute constant � .

Then has a subgraph which isvertex-disjoint �’s.

This case has no exceptional graph.

The Regularity Lemma and Applications – p.49/60

Page 161: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

A final application

We will begin with a family of graphs:

��

� �

��� ��

� � � � � � ��

We will take an arbitrary .

Then, add edges at random to , forming .

What is the diameter of the “random” graph ?

The Regularity Lemma and Applications – p.50/60

Page 162: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

A final application

We will begin with a family of graphs:

��

� �

��� ��

� � � � � � ��

We will take an arbitrary� �

� �

.

Then, add edges at random to , forming .

What is the diameter of the “random” graph ?

The Regularity Lemma and Applications – p.50/60

Page 163: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

A final application

We will begin with a family of graphs:

��

� �

��� ��

� � � � � � ��

We will take an arbitrary� �

� �

.

Then, add � edges at random to , forming .

What is the diameter of the “random” graph ?

The Regularity Lemma and Applications – p.50/60

Page 164: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

A final application

We will begin with a family of graphs:

��

� �

��� ��

� � � � � � ��

We will take an arbitrary� �

� �

.

Then, add � edges at random to , forming .

What is the diameter of the “random” graph ?

The Regularity Lemma and Applications – p.50/60

Page 165: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Many possibilities

Let’s assume

is a small constant.

could be a variety of possibilites, including

� a traditional random graph,

� an �-regular pair,

� � � �

disjoint cliques

The Regularity Lemma and Applications – p.51/60

Page 166: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Diameter 7

� Fix a constant

.

Let be an arbitrary member of .

Add edges at random to ,forming .

Let as .

Then, .

The Regularity Lemma and Applications – p.52/60

Page 167: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Diameter 7

� Fix a constant

.

� Let be an arbitrary member of

��

� �

.

Add edges at random to ,forming .

Let as .

Then, .

The Regularity Lemma and Applications – p.52/60

Page 168: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Diameter 7

� Fix a constant

.

� Let be an arbitrary member of

��

� �

.

� Add � � � ��

edges at random to ,forming .

Let as .

Then, .

The Regularity Lemma and Applications – p.52/60

Page 169: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Diameter 7

� Fix a constant

.

� Let be an arbitrary member of

��

� �

.

� Add � � � ��

edges at random to ,forming .

� Let � as � .

Then,� ��� � � � �

.

The Regularity Lemma and Applications – p.52/60

Page 170: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The algorithm

Construct � �

� � greedily such that

� ��� � � ��

�� � �

� �

Since , we stop in steps.

The Regularity Lemma and Applications – p.53/60

Page 171: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

The algorithm

Construct � �

� � greedily such that

� ��� � � ��

�� � �

� �

Since

� � ��

� � � �, we stop in� � � �

steps.

The Regularity Lemma and Applications – p.53/60

Page 172: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Quick facts

� By the partition, every vertex is in the first orsecond neighborhood of some �

� .

There is an edge in any pairbecause .

So, for vertices and , the worst case is ifthey are in second neighborhoods of different

.

The Regularity Lemma and Applications – p.54/60

Page 173: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Quick facts

� By the partition, every vertex is in the first orsecond neighborhood of some �

� .

� There is an edge in any pair� � �

� � �� �

because � ��

.

So, for vertices and , the worst case is ifthey are in second neighborhoods of different

.

The Regularity Lemma and Applications – p.54/60

Page 174: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Quick facts

� By the partition, every vertex is in the first orsecond neighborhood of some �

� .

� There is an edge in any pair� � �

� � �� �

because � ��

.

� So, for vertices � and �, the worst case is ifthey are in second neighborhoods of different

�� .

The Regularity Lemma and Applications – p.54/60

Page 175: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Diagram

First neighborhoods are red ovals.Second neighborhoods are yellow ovals.

��

� �

The Regularity Lemma and Applications – p.55/60

Page 176: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Diagram

A random edge is in

� � ��

� � �� �

.

Voilá! A path of length 7.

��

� �

The Regularity Lemma and Applications – p.55/60

Page 177: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Diagram

A random edge is in

� � ��

� � �� �

.Voilá! A path of length 7.

��

� �

The Regularity Lemma and Applications – p.55/60

Page 178: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Infinite edges

We can do better with Regularity:

THEOREM [Bohman-Frieze-Krivelevich-M., 200?]Fix a constant

. Let be an arbitrary memberof

��

� �

. Add � edges at random to , forming. If � as � , then

��� � � ��� � � � � �

as � .

The Regularity Lemma and Applications – p.56/60

Page 179: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Infinite edges

We can do better with Regularity:

THEOREM [Bohman-Frieze-Krivelevich-M., 200?]Fix a constant

. Let be an arbitrary memberof

��

� �

. Add � edges at random to , forming. If � as � , then

��� � � ��� � � � � �

as � .

The Regularity Lemma and Applications – p.56/60

Page 180: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Small world problem

In 1967, Stanley Milgram conducted a famousexperiment.

He asked 160 families in Omaha, NE, to get apackage to a stockbroker in Boston who had ahome in Sharon, MA.

It had to be mailed or carried by hand from oneacquaintance to the next, until it was delivered inperson to the broker. Before the experimentbegan, Milgram asked his colleagues how manysteps they thought it would take for the packagesto make the trip.

The Regularity Lemma and Applications – p.57/60

Page 181: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Small world problem

In 1967, Stanley Milgram conducted a famousexperiment.

He asked 160 families in Omaha, NE, to get apackage to a stockbroker in Boston who had ahome in Sharon, MA.

It had to be mailed or carried by hand from oneacquaintance to the next, until it was delivered inperson to the broker. Before the experimentbegan, Milgram asked his colleagues how manysteps they thought it would take for the packagesto make the trip.

The Regularity Lemma and Applications – p.57/60

Page 182: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Small world problem

In 1967, Stanley Milgram conducted a famousexperiment.

He asked 160 families in Omaha, NE, to get apackage to a stockbroker in Boston who had ahome in Sharon, MA.

It had to be mailed or carried by hand from oneacquaintance to the next, until it was delivered inperson to the broker. Before the experimentbegan, Milgram asked his colleagues how manysteps they thought it would take for the packagesto make the trip.

The Regularity Lemma and Applications – p.57/60

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Small world problem

The consensus was around 100.

The Regularity Lemma and Applications – p.57/60

Page 184: The Regularity Lemma and Applications - CMUrymartin/cv/reglempro.pdf · Ryan Martin rymartin@andrew.cmu.edu Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie

Small world problem

The consensus was around 100.

In fact, it took about five or six intermediarytransactions for most of the packages to get fromOmaha to the broker

The Regularity Lemma and Applications – p.57/60

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Small world problem

The consensus was around 100.

In fact, it took about five or six intermediarytransactions for most of the packages to get fromOmaha to the broker

– the "six degrees of separation" that we’refamiliar with today.

The Regularity Lemma and Applications – p.57/60

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Popular culture

We’ve seen similar phenomena in theKevin Bacon game.

The Regularity Lemma and Applications – p.58/60

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Popular culture

Also in the Erdos number project.

The Regularity Lemma and Applications – p.58/60

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Our answer

So, if familiarity grows proportionally with the sizeof the population and the amount of randomnessin the system grows also, then

It’s FIVE degrees of separation, not six!

The Regularity Lemma and Applications – p.59/60

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Our answer

So, if familiarity grows proportionally with the sizeof the population and the amount of randomnessin the system grows also, then

It’s FIVE degrees of separation, not six!

The Regularity Lemma and Applications – p.59/60

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Thanks

Thank you for letting me talk today.

The file for this talk is available online at my website:http://www.math.cmu.edu/˜rymartin

These slides were created by the Prosper document preparation system.

The Regularity Lemma and Applications – p.60/60