Ramsey numbers

Ramsey numbers

Define

R

⁢

(

a

,

b

)

to be the least integer

N

such that, in any red-blue 2-coloring of the edges of a

N

-vertex

complete graph

K

N

,
there must exist either an all-red

K

a

or an all-blue

K

b

.

Frank Ramsey proved these numbers always exist.
He famously pointed out that among any 6 people, some three are mutual friends
or some three mutual non-friends. That is,

R

⁢

(

3

,

3

)

≤

6

.
Since a red pentagon with a blue pentagram drawn inside it has no monochromatic triangle,

R

⁢

(

3

,

3

)

≥

6

. So

R

⁢

(

3

,

3

)

=

6

.

Special attention is usually paid to the diagonal

R

⁢

(

k

,

k

)

, which is often just written

R

⁢

(

k

)

.

One can also generalize this in various ways,
e.g. consider

R

⁢

(

a

,

b

,

c

)

for

three

-

colorings

of edges, etc (any number of

arguments

), and allow general sets of graphs, not just pairs of

complete

ones.

Ramsey numbers

are very difficult to determine.
To prove

lower bounds

, construct good edge-colorings of some

K

N

and, use a clique-finder
to find the largest mono-colored cliques.
To prove

upper bounds

, the main tool has been

R

⁢

(

a

,

b

)

≤

R

⁢

(

a

-

1

,

b

)

+

R

⁢

(

b

-

1

,

a

)

which implies

R

⁢

(

a

,

b

)

≤

(

a

+

b

-

2

a

-

1

)

and then

R

(

k

)

≤

[

1

+

o

(

1

)

]

4

k

/

(

4

π

⁢

k

when

k

→

∞

.
From considering random colorings and using a probabilistic nonconstructive

existence

argument, one may show

R

⁢

(

k

)

≥

k

⁢

2

k

/

2

⁢

[

o

⁢

(

1

)

+

2

/

e

]

.
It is known that

R

⁢

(

1

)

=

1

,

R

⁢

(

2

)

=

2

,

R

⁢

(

3

)

=

6

,

R

⁢

(

4

)

=

18

, and

43

≤

R

⁢

(

5

)

≤

49

.
For a survey of the best upper and lower bounds available on small
Ramsey numbers, see

http://www.

combinatorics

.org/Surveys/ds1.pdf

Radziszowski’s survey
(

http://www.cs.rit.edu/ spr/

alternate link).
Another kind of Ramsey-like number which has not gotten as much attention as it deserves,
are Ramsey numbers for

directed

graphs.
Let

R

→

⁢

(

n

)

denote the least integer

N

so that any

tournament

(complete

directed graph

with singly-directed arcs) with

≥

N

vertices contains an

acyclic

(also called “

transitive

”)

n

-node tournament. (

Analogies

: 2-color the edges

→

two directions for arcs.
Monochromatic

→

acyclic, i.e. all arcs “point one way.”)

Again, to prove lower bounds, construct good tournaments and apply something like
a clique-finder (but instead aimed at trying to find the largest acyclic

induced subgraph

).
To prove upper bounds, the main tool is

R

→

⁢

(

n

+

1

)

≤

2

⁢

R

→

⁢

(

n

)

.
That can be used to show the upper bound, and
random-orientation arguments combined with a nonconstructive probabilistic existence argument
show the lower bound, in

[

1

+

o

⁢

(

1

)

]

⁢

2

n

+

1

)

/

2

≤

R

→

⁢

(

n

)

≤

55

⋅

2

n

-

7

.
It is known that

R

→

⁢

(

1

)

=

1

,

R

→

⁢

(

2

)

=

2

,

R

→

⁢

(

3

)

=

4

,

R

→

⁢

(

4

)

=

8

,

R

→

⁢

(

5

)

=

14

,

R

→

⁢

(

6

)

=

28

,
and

32

≤

R

→

⁢

(

7

)

≤

55

.
For a full survey of directed graph Ramsey numbers includng proofs and refererences,
see

http://www.rangevoting.org/PuzzRamsey.html

Smith’s survey.

Title

Ramsey numbers

Canonical name

RamseyNumbers

Date of creation

2013-03-22 16:16:32

Last modified on

2013-03-22 16:16:32

Owner

wdsmith (13774)

Last modified by

wdsmith (13774)

Numerical id

10

Author

wdsmith (13774)

Entry type

Definition

Classification

msc 05D10

Related topic

RamseysTheorem2

Defines

Ramsey numbers

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