Ramsey’s theorem

Ramsey’s theorem

The original version of

Ramsey’s

theorem

states that for
every

positive integers

k

1

and

k

2

there is

n

such that if
edges of a

complete graph

on

n

vertices are colored in two
colors, then there is either a

k

1

-clique of the first color or

k

2

-clique of the second color.

The standard proof proceeds by

induction

on

k

1

+

k

2

. If

k

1

,

k

2

≤

2

, then the theorem holds trivially. To prove
induction step we consider the graph

G

that contains no cliques
of the desired kind, and then consider any vertex

v

of

G

.

Partition

the rest of the vertices into two classes

C

1

and

C

2

according to whether edges from

v

are of color

1

or

2

respectively. By

inductive hypothesis

|

C

1

|

is

bounded

since
it contains no

k

1

-

1

-clique of color

1

and no

k

2

-clique of
color

2

. Similarly,

|

C

2

|

is bounded. QED.

Similar

argument

shows that for any positive integers

k

1

,

k

2

,

…

,

k

t

if we color the edges of a sufficiently large
graph in

t

colors, then we would be able to find either

k

1

-clique of color

1

, or

k

2

-clique or color

2

,…, or

k

t

-clique of color

t

.

The minimum

n

whose

existence

stated in Ramsey’s theorem is
called

Ramsey number

and denoted by

R

⁢

(

k

1

,

k

2

)

(and

R

⁢

(

k

1

,

k

2

,

…

,

k

t

)

for multicolored graphs). The above proof
shows that

R

⁢

(

k

1

,

k

2

)

≤

R

⁢

(

k

1

,

k

2

-

1

)

+

R

⁢

(

k

1

-

1

,

k

2

)

. From that
it is not hard to deduce by induction that

R

⁢

(

k

1

,

k

2

)

≤

(

k

1

+

k

2

-

2

k

1

-

1

)

. In the most interesting case

k

1

=

k

2

=

k

this yields approximately

R

⁢

(

k

,

k

)

≤

(

4

+

o

⁢

(

1

)

)

k

. The
lower bounds can be established by means of probabilistic
construction as follows.

Take a complete graph on

n

vertices and color its edges at random,
choosing the color of each edge uniformly independently of all other edges.
The

probability

that any given set of

k

vertices is a
monochromatic clique is

2

1

-

k

. Let

I

r

be the

random
variable

which is

1

if

r

’th set of

k

elements

is
monochromatic clique and is

0

otherwise. The sum

I

r

’s
over all

k

-element sets is
simply the number of monochromatic

k

-cliques. Therefore by
linearity of

expectation

E

⁢

(

∑

I

r

)

=

∑

E

⁢

(

I

r

)

=

2

1

-

k

⁢

(

n

k

)

. If the expectation is less than

1

,
then there exists a

coloring

which has no monochromatic cliques. A
little exercise in calculus shows that if we choose

n

to be no more
than

(

2

+

o

⁢

(

1

)

)

k

/

2

then the expectation is indeed less than

1

. Hence,

R

⁢

(

k

,

k

)

≥

(

2

+

o

⁢

(

1

)

)

k

.

The gap between the lower and upper bounds has been open (

http://planetmath.org/Open3

) for
several decades. There have been a number of improvements in

o

⁢

(

1

)

terms, but nothing better than

2

+

o

⁢

(

1

)

≤

R

⁢

(

k

,

k

)

1

/

k

≤

4

+

o

⁢

(

1

)

is known. It is not even known whether

lim

k

→

∞

⁡

R

⁢

(

k

,

k

)

1

/

k

exists.

The behavior of

R

⁢

(

k

,

x

)

for fixed

k

and large

x

is equally
mysterious. For this case Ajtai, Komlós and
Szemerédi

[

1

]

proved that

R

⁢

(

k

,

x

)

≤

c

k

⁢

x

k

-

1

/

(

ln

)

k

-

2

. The

matching

lower bound has only
recently been established for

k

=

3

by Kim

[

4

]

. Even in this case the asymptotics is
unknown. The

combination

of results of Kim and improvement of
Ajtai, Komlós and Szemerédi’s result by Shearer

[

6

]

yields

1

162

⁢

(

1

+

o

⁢

(

1

)

)

⁢

k

2

/

log

⁡

k

≤

R

⁢

(

3

,

k

)

≤

(

1

+

o

⁢

(

1

)

)

⁢

k

2

/

log

⁡

k

.

A lot of

machine

and human time has been spent trying to
determine Ramsey numbers for small

k

1

and

k

2

. An up-to-date
summary of our knowledge about small Ramsey numbers can be found
in

[

5

]

.

References

1

Miklós Ajtai, János Komlós, and Endre Szemerédi.

A note on Ramsey numbers.

J. Combin. Theory Ser. A

, 29(3):354–360, 1980.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0455.05045

Zbl 0455.05045.

2

Noga Alon and Joel H. Spencer.

The

probabilistic method

.

John Wiley & Sons, Inc., second edition, 2000.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0996.05001

Zbl 0996.05001.

3

Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer.

Ramsey Theory

.

Wiley-Interscience series in

discrete mathematics

. 1980.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0455.05002

Zbl 0455.05002.

4

Jeong Han Kim.

The Ramsey number

R

⁢

(

3

,

t

)

has order of magnitude

t

2

/

log

⁡

t

.

Random

Structures

& Algorithms

, 7(3):173–207, 1995.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0832.05084

Zbl 0832.05084.

Preprint is

http://research.microsoft.com/research/theory/jehkim/papers/ramsey5.pdf

available online.

5

Stanislaw Radziszowski.

Small Ramsey numbers.

Electronic Journal of Combinatorics

, Dynamical Survey, 2002.

http://www.

combinatorics

.org/Surveys/ds1.pdf

Available
online.

6

James B. Shearer.

A note on the

independence number

of triangle-free graphs.

Discrete Mathematics

, 46:83–87, 1983.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0516.05053

Zbl 0516.05053.

Abstract is

http://www.research.ibm.com/people/s/shearer/abstracts/intfg.html

available online

Title

Ramsey’s theorem

Canonical name

RamseysTheorem

Date of creation

2013-03-22 13:53:09

Last modified on

2013-03-22 13:53:09

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

18

Author

bbukh (348)

Entry type

Theorem

Classification

msc 05D40

Classification

msc 05D10

Related topic

RamseysTheorem

Related topic

ProbabilisticMethod

Defines

Ramsey number

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