aarjaneiro/planetmath huggingface dataset:
null graph

null graph

The

null graph

is the graph with no vertices or edges. The null graph is the

initial object

in the

category of graphs

.

Further Reading

•

Harary, F. and Read, R. “Is the null-graph a pointless concept?”, Lecture Notes in Mathematics 406 (1974), pp. 37-44.

Title

null graph

Canonical name

NullGraph

Date of creation

2013-03-22 12:48:42

Last modified on

2013-03-22 12:48:42

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 05C99

Synonym

empty graph

Generated on Fri Feb 9 07:57:31 2018 by

LaTeXML
order (of a graph)

order (of a graph)

The

order

of a graph

G

is the number of vertices in

G

; it is denoted by

|

G

|

. The same notation is used for the number of elements (cardinality) of a set. Thus,

|

G

|

=

|

V

⁢

(

G

)

|

. We write

G

n

for an

arbitrary graph of order n

. Similarly,

G

⁢

(

n

,

m

)

denotes an

arbitrary graph of order n and size m

.

Adapted with permission of the author from

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

order (of a graph)

Canonical

name

OrderofAGraph

Date of creation

2013-03-22 12:31:23

Last modified on

2013-03-22 12:31:23

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05C99

Synonym

order

Related topic

Graph

Related topic

SizeOfAGraph

Related topic

MantelsTheorem

Generated on Fri Feb 9 07:57:33 2018 by

LaTeXML
Poincaré formula

Poincaré formula

Let

K

be finite oriented

simplicial complex

of

dimension

n

. Then

χ

⁢

(

K

)

=

∑

p

=

0

n

(

-

1

)

p

⁢

R

p

⁢

(

K

)

,

where

χ

⁢

(

K

)

is the

Euler characteristic

of

K

, and

R

p

⁢

(

K

)

is the

p

-th

Betti number

of

K

.

This formula also works when

K

is any finite CW complex. The Poincaré formula is also known as the Euler-Poincaré formula, for it is a generalization of the

Euler formula

for polyhedra.

If

K

is a

compact

connected

orientable surface with no

boundary

and with genus h, then

χ

⁢

(

K

)

=

2

-

2

⁢

h

. If

K

is non-orientable instead, then

χ

⁢

(

K

)

=

2

-

h

.

Title

Poincaré formula

Canonical name

PoincareFormula

Date of creation

2013-03-22 13:40:15

Last modified on

2013-03-22 13:40:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 05C99

Synonym

Euler-Poincaré formula

Synonym

Euler-Poincare formula

Related topic

EulersPolyhedronTheorem

Related topic

Polytope

Generated on Fri Feb 9 07:57:35 2018 by

LaTeXML
proof of Turan’s theorem

proof of Turan’s theorem

If the graph

G

has

n

≤

p

-

1

vertices it cannot contain any

p

–clique and thus has at most

(

n

2

)

edges. So in this case we only have to prove that

n

⁢

(

n

-

1

)

2

≤

(

1

-

1

p

-

1

)

⁢

n

2

2

.

Dividing by

n

2

we get

n

-

1

n

=

1

-

1

n

≤

1

-

1

p

-

1

,

which is true since

n

≤

p

-

1

.

Now we assume that

n

≥

p

and the set of vertices of

G

is denoted by

V

. If

G

has the maximum number of edges possible without containing a

p

–clique it contains a

p

-

1

–clique, since otherwise we might add edges to get one. So we denote one such clique by

A

and define

B

:=

G

\

A

.

So

A

has

(

p

-

1

2

)

edges. We are now interested in the number of edges in

B

, which we will call

e

B

, and in the number of edges connecting

A

and

B

, which will be called

e

A

,

B

. By

induction

we get:

e

B

≤

1

2

⁢

(

1

-

1

p

-

1

)

⁢

(

n

-

p

+

1

)

2

.

Since

G

does not contain any

p

–clique every vertice of

B

is

connected

to at most

p

-

2

vertices in

A

and thus we get:

e

A

,

B

≤

(

p

-

2

)

⁢

(

n

-

p

+

1

)

.

Putting this together we get for the number of edges

|

E

|

of

G

:

|

E

|

≤

(

p

-

1

2

)

+

1

2

⁢

(

1

-

1

p

-

1

)

⁢

(

n

-

p

+

1

)

2

+

(

p

-

2

)

⁢

(

n

-

p

+

1

)

.

And thus we get:

|

E

|

≤

(

1

-

1

p

-

1

)

⁢

n

2

2

.

Title

proof of Turan’s

theorem

Canonical name

ProofOfTuransTheorem

Date of creation

2013-03-22 12:46:13

Last modified on

2013-03-22 12:46:13

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

6

Author

mathwizard (128)

Entry type

Proof

Classification

msc 05C99

Generated on Fri Feb 9 07:57:37 2018 by

LaTeXML
realization of a graph

realization of a graph

A

realization of a graph

G

in a

topological space

X

is an injective map

ρ

:

G

→

X

such that

G

with the

graph topology

is

homeomorphic

to

ρ

⁢

(

G

)

with the

subspace topology

. Of particular interest are realizations of

G

in 3-dimensional

Euclidean space

, where each of the edges is a

line segment

, and where no 4 points of

G

are

coplanar

.

Title

realization of a graph

Canonical name

RealizationOfAGraph

Date of creation

2013-03-22 12:31:58

Last modified on

2013-03-22 12:31:58

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

7

Author

rmilson (146)

Entry type

Definition

Classification

msc 05C99

Related topic

Subdivision

Related topic

GraphTopology

Generated on Fri Feb 9 07:57:41 2018 by

LaTeXML
size (of a graph)

size (of a graph)

The

size

of a graph

G

is the number of edges in

G

; it is denoted by

e

⁢

(

G

)

.

G

⁢

(

n

,

m

)

denotes an

arbitrary graph of order n and size m

.

Adapted with permission of the author from

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

size (of a graph)

Canonical

name

SizeofAGraph

Date of creation

2013-03-22 12:31:19

Last modified on

2013-03-22 12:31:19

Owner

mps (409)

Last modified by

mps (409)

Numerical id

8

Author

mps (409)

Entry type

Definition

Classification

msc 05C99

Synonym

size

Related topic

Graph

Related topic

OrderOfAGraph

Related topic

MantelsTheorem

Generated on Fri Feb 9 07:57:43 2018 by

LaTeXML
subdivision

subdivision

A graph

H

is said to be a

subdivision

, or

topological minor

of a graph

G

, or a

topological

G

graph

if

H

is obtained from

G

by subdividing some of the edges, that is, by replacing the edges by paths having at most their

endvertices

in common. We often use

T

⁢

G

for a topological

G

graph.

Thus,

T

⁢

G

denotes

any

member of a large family of graphs; for example,

T

⁢

C

4

is an arbitrary cycle of length at least 4. For any graph

G

, the spaces

R

⁢

(

G

)

(denoting the realization of G) and

R

⁢

(

T

⁢

G

)

are homeomorphic.

Adapted with permission of the author from

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

subdivision

Canonical

name

Subdivision

Date of creation

2013-03-22 12:31:51

Last modified on

2013-03-22 12:31:51

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05C99

Synonym

topological minor

Related topic

Homeomorphic

Related topic

Realization

Generated on Fri Feb 9 07:57:45 2018 by

LaTeXML
subgraph

subgraph

We say that

G

′

=

(

V

′

,

E

′

)

is a

subgraph

of

G

=

(

V

,

E

)

if

V

′

⊆

V

and

E

′

⊆

E

. In this case we write

G

′

⊆

G

.

If

G

′

contains

all edges

of

G

that join two vertices in

V

′

then

G

′

is said to be the subgraph

induced

or

spanned by

V

′

and is denoted by

G

⁢

[

V

′

]

. Thus, a subgraph

G

′

of

G

is an

induced subgraph

if

G

′

=

G

⁢

[

V

⁢

(

G

′

)

]

. If

V

′

=

V

, then

G

′

is said to be a

spanning

subgraph of

G

.

Often, new graphs are constructed from old ones by deleting or adding some vertices and edges. If

W

⊂

V

⁢

(

G

)

, then

G

-

W

=

G

⁢

[

V

∖

W

]

is the subgraph of

G

obtained by deleting the vertices in

W

and all edges

incident

with them

. Similarly, if

E

′

⊆

E

⁢

(

G

)

, then

G

-

E

′

=

(

V

⁢

(

G

)

,

E

⁢

(

G

)

∖

E

′

)

. If

W

=

w

and

E

′

=

x

⁢

y

, then this notation is simplified to

G

-

w

and

G

-

x

⁢

y

. Similarly, if

x

and

y

are nonadjacent vertices of

G

, then

G

+

x

⁢

y

is obtained from

G

by joining

x

to

y

.

Adapted with permission of the author from

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

subgraph

Canonical

name

Subgraph

Date of creation

2013-03-22 12:30:59

Last modified on

2013-03-22 12:30:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05C99

Related topic

Graph

Related topic

Pseudograph

Related topic

Multigraph

Defines

induced

Defines

spanned by

Defines

spanning

Defines

spanning subgraph

Defines

induced subgraph

Generated on Fri Feb 9 07:57:47 2018 by

LaTeXML
Turan’s theorem

Turan’s theorem

A graph with

n

vertices, which contains no

p

-

http://planetmath.org/node/1757

clique with

p

≥

2

, has at most

(

1

-

1

p

-

1

)

⁢

n

2

2

edges.

Title

Turan’s theorem

Canonical name

TuransTheorem

Date of creation

2013-03-22 12:44:10

Last modified on

2013-03-22 12:44:10

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

5

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 05C99

Generated on Fri Feb 9 07:57:49 2018 by

LaTeXML
uniformly locally finite graph

uniformly locally finite graph

A is a

locally finite graph

(

http://planetmath.org/LocallyFiniteGraph

)

(

V

,

E

)

such that there exists an

M

∈

ℕ

such that for every

x

∈

V

we have that the degree of

x

, also denoted

ρ

⁢

(

x

)

, is at most

M

. In other words there exists an

M

∈

ℕ

such that for every

x

∈

V

,

ρ

⁢

(

x

)

≤

M

.

Note that the examples provided in locally finite graph (

http://planetmath.org/LocallyFiniteGraph

) are also examples of a

uniformly locally finite graph

since both graphs are

regular

(

http://planetmath.org/RegularGraph

) and have finite degree (

http://planetmath.org/Degree7

) at each vertex.

Title

uniformly locally finite graph

Canonical

name

UniformlyLocallyFiniteGraph

Date of creation

2013-03-22 16:00:54

Last modified on

2013-03-22 16:00:54

Owner

sjm1979 (13837)

Last modified by

sjm1979 (13837)

Numerical id

6

Author

sjm1979 (13837)

Entry type

Definition

Classification

msc 05C99

Related topic

LocallyFiniteGraph

Generated on Fri Feb 9 07:57:51 2018 by

LaTeXML
Wagner’s theorem

Wagner’s theorem

Theorem 1

(Wagner)

A graph is planar if and only if it contains neither

K

5

nor

K

3

,

3

as a minor, where

K

5

is the

complete graph

of order 5 and

K

3

,

3

is the

complete bipartite graph

of order 6.

Wagner’s theorem is to Kuratowski’s theorem.

Title

Wagner’s theorem

Canonical name

WagnersTheorem

Date of creation

2013-03-22 12:31:49

Last modified on

2013-03-22 12:31:49

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 05C99

Related topic

PlanarGraph

Related topic

KuratowskisTheorem

Generated on Fri Feb 9 07:57:52 2018 by

LaTeXML
wheel graph

wheel graph

The

wheel graph

of

n

vertices

W

n

is a graph that contains a cycle of length

n

-

1

plus a vertex

v

(sometimes called the

hub

) not in the cycle such that

v

is

connected

to every other vertex. The edges connecting

v

to the rest of the graph are sometimes called

spokes

.

W

4

:

\xymatrix

⁢

&

⁢

A

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

d

⁢

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

d

⁢

r

]

⁢

&

⁢

&

⁢

D

⁢

&

⁢

C

⁢

\ar

⁢

@

-

[

u

⁢

r

]

⁢

\ar

⁢

@

-

[

r

⁢

r

]

⁢

&

⁢

&

⁢

B

⁢

\ar

⁢

@

-

[

u

⁢

l

]

W

6

:

\xymatrix

⁢

&

⁢

&

⁢

A

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

d

⁢

r

⁢

r

]

⁢

&

⁢

&

⁢

E

⁢

\ar

⁢

@

-

[

r

⁢

r

]

⁢

\ar

⁢

@

-

[

u

⁢

r

⁢

r

]

⁢

&

⁢

&

⁢

F

⁢

&

⁢

&

⁢

B

⁢

\ar

⁢

@

-

[

l

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

D

⁢

\ar

⁢

@

-

[

u

⁢

r

]

⁢

\ar

⁢

@

-

[

u

⁢

l

]

⁢

&

⁢

&

⁢

C

⁢

\ar

⁢

@

-

[

u

⁢

l

]

⁢

\ar

⁢

@

-

[

l

⁢

l

]

⁢

&

Title

wheel graph

Canonical

name

WheelGraph

Date of creation

2013-03-22 12:17:27

Last modified on

2013-03-22 12:17:27

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

5

Author

vampyr (22)

Entry type

Definition

Classification

msc 05C99

Synonym

hub

Synonym

spoke

Generated on Fri Feb 9 07:57:54 2018 by

LaTeXML
LYM inequality

LYM inequality

Let

ℱ

be a

Sperner family

, that is, the

collection

of
subsets of

{

1

,

2

,

…

,

n

}

such that no set contains any other subset.
Then

∑

X

∈

ℱ

1

(

n

|

X

|

)

≤

1

.

This inequality is known as

LYM inequality

by the names of
three people that independently discovered it:
Lubell

[

2

]

, Yamamoto

[

4

]

,
Meshalkin

[

3

]

.

Since

(

n

k

)

≤

(

n

⌊

n

/

2

⌋

)

for every integer

k

, LYM inequality tells us that

|

F

|

/

(

n

⌊

n

/

2

⌋

)

≤

1

which is Sperner’s

theorem

(

http://planetmath.org/SpernersTheorem

).

References

1

Konrad Engel.

Sperner theory

, volume 65 of

Encyclopedia of Mathematics
and Its Applications

.

Cambridge University Press.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0868.05001

Zbl 0868.05001.

2

David Lubell.

A short proof of Sperner’s lemma.

J. Comb. Theory

, 1:299, 1966.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0151.01503

Zbl 0151.01503.

3

Lev D. Meshalkin.

Generalization

of Sperner’s theorem on the number of subsets of a

finite set

.

Teor. Veroyatn. Primen.

, 8:219–220, 1963.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0123.36303

Zbl 0123.36303.

4

Koichi Yamamoto.

Logarithmic order of free

distributive lattice

.

J. Math. Soc. Japan

, 6:343–353, 1954.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0056.26301

Zbl 0056.26301.

Title

LYM inequality

Canonical name

LYMInequality

Date of creation

2013-03-22 14:06:01

Last modified on

2013-03-22 14:06:01

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Theorem

Classification

msc 05D05

Classification

msc 06A07

Synonym

LYM-inequality

Related topic

SpernersTheorem

Generated on Fri Feb 9 07:57:56 2018 by

LaTeXML
Sperner’s theorem

Sperner’s theorem

What is the size of the largest family

ℱ

of subsets of
an

n

-

element

set such that no

A

∈

ℱ

is a subset of

B

∈

ℱ

? Sperner

[

3

]

gave an
answer in the following elegant

theorem

:

Theorem

.

For every family

F

of

incomparable

subsets of an

n

-set,

|

F

|

≤

(

n

⌊

n

/

2

⌋

)

.

A family satisfying the conditions of Sperner’s theorem is usually
called

Sperner family

or

antichain

. The later
terminology stems from the fact that subsets of a

finite set

ordered by

inclusion

form a

Boolean lattice

.

There are many

generalizations

of Sperner’s theorem. On one hand,
there are

refinements

like

LYM inequality

that strengthen the
theorem in various ways. On the other hand, there are
generalizations to posets other than the Boolean lattice. For a
comprehensive exposition of the topic one should consult a
well-written monograph by Engel

[

2

]

.

References

1

Béla Bollobás.

Combinatorics

: Set systems,

hypergraphs

, families of vectors, and
combinatorial probability.

1986.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0595.05001

Zbl
0595.05001.

2

Konrad Engel.

Sperner theory

, volume 65 of

Encyclopedia of Mathematics
and Its Applications

.

Cambridge University Press.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0868.05001

Zbl
0868.05001.

3

Emanuel Sperner.

Ein Satz über Untermengen einer endlichen Menge.

Math. Z.

, 27:544–548, 1928.

http://www.emis.de/cgi-bin/jfmen/MATH/JFM/quick.html?first=1&maxdocs=20&type=html&an=JFM%2054.0090.06&format=

complete

Available online at

http://www.emis.de/projects/JFM/

JFM.

Title

Sperner’s theorem

Canonical name

SpernersTheorem

Date of creation

2013-03-22 13:51:57

Last modified on

2013-03-22 13:51:57

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

7

Author

bbukh (348)

Entry type

Theorem

Classification

msc 05D05

Classification

msc 06A07

Related topic

LYMInequality

Defines

Sperner family

Generated on Fri Feb 9 07:57:58 2018 by

LaTeXML
coloring

coloring

A

coloring

of a set

X

by

Y

is just a

function

f

:

X

→

Y

. The term coloring is used because the function can be thought of as assigning a “color” from

Y

to each

element of

X

.

Any coloring provides a

partition

of

X

: for each

y

∈

Y

,

f

-

1

⁢

(

y

)

, the set of

elements

x

such that

f

⁢

(

x

)

=

y

, is one element of the partition. Since

f

is a function, the sets in the partition are

disjoint

, and since it is a

total function

, their union is

X

.

Title

coloring

Canonical name

Coloring

Date of creation

2013-03-22 12:55:43

Last modified on

2013-03-22 12:55:43

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 05D10

Synonym

colouring

Related topic

Partition

Related topic

GraphTheory

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Erdős-Rado theorem

Erdős-Rado theorem

Repeated

exponentiation

for

cardinals

is denoted

exp

i

⁡

(

κ

)

, where

i

<

ω

. It is defined by:

exp

0

⁡

(

κ

)

=

κ

and

exp

i

+

1

⁡

(

κ

)

=

2

exp

i

⁡

(

κ

)

The Erdős-Rado

theorem

states that:

exp

i

(

κ

)

+

→

(

κ

+

)

κ

i

+

1

That is, if

f

:

[

exp

i

(

κ

)

+

]

i

+

1

→

κ

then there is a

homogeneous set

of size

κ

+

.

As special cases,

(

2

κ

)

+

→

(

κ

+

)

κ

2

and

(

2

ℵ

0

)

+

→

(

ℵ

1

)

ℵ

0

2

.

Title

Erdős-Rado theorem

Canonical name

ErdHosRadoTheorem

Date of creation

2013-03-22 12:59:30

Last modified on

2013-03-22 12:59:30

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

9

Author

Henry (455)

Entry type

Theorem

Classification

msc 05D10

Classification

msc 03E05

Related topic

arrowsrelation

Related topic

ArrowsRelation

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proof of Ramsey’s theorem

proof of Ramsey’s theorem

ω

→

(

ω

)

k

n

is proven by

induction

on

n

.

If

n

=

1

then this just states that any

partition

of an

infinite set

into a finite number of subsets must include an infinite set; that is, the union of a finite number of

finite sets

is finite. This is simple enough to prove: since there are a finite number of sets, there is a largest set of size

x

. Let the number of sets be

y

. Then the size of the union is no more than

x

⁢

y

.

If

ω

→

(

ω

)

k

n

then we can show that

ω

→

(

ω

)

k

n

+

1

Let

f

be some

coloring

of

[

S

]

n

+

1

by

k

where

S

is an infinite subset of

ω

. Observe that, given an

x

<

ω

, we can define

f

x

:

[

S

∖

{

x

}

]

n

→

k

by

f

x

⁢

(

X

)

=

f

⁢

(

{

x

}

∪

X

)

. Since

S

is

infinite

, by the

induction hypothesis

this will have an infinite

homogeneous set

.

Then we define a

sequence

of integers

⟨

n

i

⟩

i

∈

ω

and a sequence of infinite subsets of

ω

,

⟨

S

i

⟩

i

∈

ω

by induction. Let

n

0

=

0

and let

S

0

=

ω

. Given

n

i

and

S

i

for

i

≤

j

we can define

S

j

as an infinite homogeneous set for

f

n

i

:

[

S

j

-

1

]

n

→

k

and

n

j

as the least element of

S

j

.

Obviously

N

=

⋃

{

n

i

}

is infinite, and it is also

homogeneous

, since each

n

i

is

contained

in

S

j

for each

j

≤

i

.

Title

proof of Ramsey’s

theorem

Canonical

name

ProofOfRamseysTheorem

Date of creation

2013-03-22 12:55:52

Last modified on

2013-03-22 12:55:52

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Proof

Classification

msc 05D10

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Ramsey’s theorem

Ramsey’s theorem

Ramsey’s theorem

states that a particular

arrows relation

,

ω

→

(

ω

)

k

n

holds for any

integers

n

and

k

.

In words, if

f

is a

function

on sets of integers of size

n

whose range is finite then there is some infinite

X

⊆

ω

such that

f

is constant on the subsets of

X

of size

n

.

As an example, consider the case where

n

=

k

=

2

, and

f

:

[

ω

]

2

→

{

0

,

1

}

is defined by:

f

⁢

(

{

x

,

y

}

)

=

{

1

if

⁢

x

=

y

2

⁢

or

⁢

y

=

x

2

0

otherwise

Then let

X

⊆

ω

be the set of integers which are not

perfect squares

. This is clearly infinite, and obviously if

x

,

y

∈

X

then neither

x

=

y

2

nor

y

=

x

2

, so

f

is

homogeneous

on

X

.

Title

Ramsey’s theorem

Canonical name

RamseysTheorem

Date of creation

2013-03-22 12:55:49

Last modified on

2013-03-22 12:55:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 05D10

Related topic

RamseysTheorem2

Related topic

GraphTheory

Generated on Fri Feb 9 07:58:11 2018 by

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Ramsey-theoretic proof of the Erdős-Szekeres theorem

Ramsey-theoretic proof of the Erdős-Szekeres theorem

Let

n

≥

3

be an

integer

. By the finite

Ramsey theorem

, there is a

positive integer

N

such that the

arrows relation

N

→

(

n

)

2

3

holds. Let

X

be a planar point set in

general position

with

|

X

|

≥

N

. Define a red-blue

colouring

of the

triangles

in

X

as follows. Let

T

=

{

a

,

b

,

c

}

be a triangle of

X

with

a

,

b

,

c

having monotonically increasing

x

-

coordinates

. (If two points have the same

x

-coordinate, break the tie by placing the point with smaller

y

-coordinate first.) If

b

lies above the line determined by

a

and

c

(the triangle “points up”), then colour the triangle blue. Otherwise,

b

lies below the line (the triangle “points down”); in this case colour the triangle red.

Now there must be a homogeneous subset

Y

⊂

X

with

|

Y

|

≥

n

. Without loss of generality, every triangle in

Y

is coloured blue. To see that this implies that the points of

Y

are the vertices of a convex

n

-gon, suppose there exist

a

,

b

,

c

,

d

in

Y

such that

d

∈

conv

⁡

{

a

,

b

,

c

}

and such that

a

,

b

,

c

have monotonically increasing

x

-coordinates (breaking ties as before). Since every triangle in

Y

is coloured blue, the triangle

{

a

,

b

,

c

}

points up. If the

x

-coordinate of

d

is less than or equal to that of

b

, then the triangle

{

a

,

d

,

b

}

points down. But if the

x

-coordinate of

d

is greater than that of

b

, the triangle

{

b

,

d

,

c

}

points down. In either case there is a red triangle in the homogeneously blue

Y

, a

contradiction

. Hence

Y

is a convex

n

-gon. This shows that

g

⁢

(

n

)

≤

N

<

∞

.

References

1

P. Erdős and G. Szekeres, A combinatorial problem in

geometry

,

Compositio Math.

2

(1935), 463–470.

2

W. Morris and V. Soltan, The Erdős–Szekeres problem on points in convex position – a survey,

Bull. Amer. Math. Soc.

37

(2000), 437–458.

Title

Ramsey-theoretic proof of the Erdős-Szekeres

theorem

Canonical name

RamseytheoreticProofOfTheErdHosSzekeresTheorem

Date of creation

2013-03-22 16:16:46

Last modified on

2013-03-22 16:16:46

Owner

mps (409)

Last modified by

mps (409)

Numerical id

5

Author

mps (409)

Entry type

Proof

Classification

msc 05D10

Classification

msc 51D20

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