Generalization of Ramsey Number

1. Generalization of Ramsey Numbers
Further Research on Ramsey Theorem
Asaad, Al-Ahmadgaid B.
email: alstated@gmail.com
Mindanao State University
Iligan Institute of Technology

2. Generalization
The definition of the Ramsey number R(p, q) with 2 parameters can be
generalized in a natural way to the Ramsey number R(p1 , p2 , · · · , pk )
with k parameters as follows. Let k, p1 , p2 , · · · , pk N with k ≥ 3.
The Ramsey number R(p1 , p2 , · · · , pk ) is the smallest natural number
n such that for any colouring of the edges of an n-clique by k colours:
colour 1, colour 2, · · · , colour k, there exist a colour i (i = 1, 2, · · · , k)
and a pi -clique in the resulting configuration such that all edges in the
pi -clique are coloured by colour i.
• The result of Example 3.4.1. shows that R(3, 3, 3) ≤ 17.
• In 1995, Greenwood and Gleason [GG] proved by construction that
R(3, 3, 3) ≥ 17.
• Thus, R(3, 3, 3) = 17
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3. Generalization
The definition of the Ramsey number R(p, q) with 2 parameters can be
generalized in a natural way to the Ramsey number R(p1 , p2 , · · · , pk )
with k parameters as follows. Let k, p1 , p2 , · · · , pk N with k ≥ 3.
The Ramsey number R(p1 , p2 , · · · , pk ) is the smallest natural number
n such that for any colouring of the edges of an n-clique by k colours:
colour 1, colour 2, · · · , colour k, there exist a colour i (i = 1, 2, · · · , k)
and a pi -clique in the resulting configuration such that all edges in the
pi -clique are coloured by colour i.
• The result of Example 3.4.1. shows that R(3, 3, 3) ≤ 17.
• In 1995, Greenwood and Gleason [GG] proved by construction that
R(3, 3, 3) ≥ 17.
• Thus, R(3, 3, 3) = 17
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4. Generalization
The definition of the Ramsey number R(p, q) with 2 parameters can be
generalized in a natural way to the Ramsey number R(p1 , p2 , · · · , pk )
with k parameters as follows. Let k, p1 , p2 , · · · , pk N with k ≥ 3.
The Ramsey number R(p1 , p2 , · · · , pk ) is the smallest natural number
n such that for any colouring of the edges of an n-clique by k colours:
colour 1, colour 2, · · · , colour k, there exist a colour i (i = 1, 2, · · · , k)
and a pi -clique in the resulting configuration such that all edges in the
pi -clique are coloured by colour i.
• The result of Example 3.4.1. shows that R(3, 3, 3) ≤ 17.
• In 1995, Greenwood and Gleason [GG] proved by construction that
R(3, 3, 3) ≥ 17.
• Thus, R(3, 3, 3) = 17
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5. Example
From Exercise 42. Show that R(3, 3, 2) = 6
Proof. Let v0 be a vertex from any of the six vertices. Joining the
vertex v0 to the vertices that incident to it, we have
v0
v1
v2
v3
v4
v5
If we are going to colour the edges with either red, blue, or yellow.
Then, by Pigeonhole principle, there are at least 3 edges coloured with
either of the three colours.
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6. Assuming the edges were {(v0 ,v1 ), (v0 ,v2 ), and (v0 ,v3 )} and that it is
coloured with red. Then
v0
v1
v2
v3
If any of the edges between v1 , v2 , and v3 is coloured with red, then
we can form a monochromatic triangle of colour red.
v0
v1
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v2
v3

7. If none of them is coloured with red, then either blue or yellow is
the colour of the edges. Suppose the colour of the edge between the
vertices v1 and v2 is blue, and the edge between the vertices v2 and v3
is yellow. That is,
v0
v1
v2
v3
Then, we can form a single edge coloured with blue, or an edge
coloured with yellow.
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8. Suppose, the edges between the vertices v1 , v2 , and v3 are coloured
with blue. If the edge formed between vertices v1 and v3 is coloured
with red. Then, we have a single monochromatic triangle coloured
with red. But, if the edge is coloured with blue, then we can have
a monochromatic triangle of colour blue. In addition, if the edge is
coloured with yellow, then we have a single line (edge) coloured with
yellow.
v0
v1
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v2
v0
v3
v1
v2
v3

9. v0
v1
v2
v3
Hence, using 6 vertices is enough to show that there exist 3-clique of
blue, 3-clique of red, or 2-clique of yellow.
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10. Applications of Ramsey Numbers
These are the areas where we can apply Ramsey numbers
• Number Theory
• Geometry
• Communications
• Decision Making
• Computer Science
http://www.cs.umd.edu/~gasarch/ramsey/ramsey.html
• and more...
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11. Applications of Ramsey Numbers
These are the areas where we can apply Ramsey numbers
• Number Theory
• Geometry
• Communications
• Decision Making
• Computer Science
http://www.cs.umd.edu/~gasarch/ramsey/ramsey.html
• and more...
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12. Applications of Ramsey Numbers
These are the areas where we can apply Ramsey numbers
• Number Theory
• Geometry
• Communications
• Decision Making
• Computer Science
http://www.cs.umd.edu/~gasarch/ramsey/ramsey.html
• and more...
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13. Applications of Ramsey Numbers
These are the areas where we can apply Ramsey numbers
• Number Theory
• Geometry
• Communications
• Decision Making
• Computer Science
http://www.cs.umd.edu/~gasarch/ramsey/ramsey.html
• and more...
8 of 18

14. Applications of Ramsey Numbers
These are the areas where we can apply Ramsey numbers
• Number Theory
• Geometry
• Communications
• Decision Making
• Computer Science
http://www.cs.umd.edu/~gasarch/ramsey/ramsey.html
• and more...
8 of 18

15. Applications of Ramsey Numbers
These are the areas where we can apply Ramsey numbers
• Number Theory
• Geometry
• Communications
• Decision Making
• Computer Science
http://www.cs.umd.edu/~gasarch/ramsey/ramsey.html
• and more...
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16. 9 of 18

17. Ramsey Theorem
In combinatorics, Ramsey’s theorem states that in any colouring of the
edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs. For two colours, Ramsey’s theorem states
that for any pair of positive integers (p, q), there exists a least positive
integer R(p, q) such that for any complete graph on R(p, q) vertices,
whose edges are coloured red or blue, there exists either a complete
subgraph on p vertices which is entirely blue, or a complete subgraph
on q vertices which is entirely red.
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18. More on Theorem
• Extension of the Theorem
The theorem can also be extended to hypergraphs. An m-hypergraph
is a graph whose ”edges” are sets of m vertices - in a normal graph
an edge is a set of 2 vertices.
• Infinite Ramsey theorem
A further result, also commonly called Ramsey’s theorem, applies
to infinite graphs. In a context where finite graphs are also being
discussed it is often called the ”Infinite Ramsey theorem”.
Theorem: Let X be some countably infinite set and colour the
elements of X (n) (the subsets of X of size n) in c different colours.
Then there exists some infinite subset M of X such that the size n
subsets of M all have the same colour.
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19. List of Ramsey Numbers
List of Ramsey numbers for p-clique and q-clique, p, q ≤ 19
p
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
q
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
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R(p, q)
6
9
14
18
23
28
36
[40, 43]
[46, 51]
[52, 59]
[59, 69]
[66, 78]
[73, 88]
[79, 135]
[92, 152]
[98, 170]
[106, 189]
Reference
Greenwood and Gleason 1955
Greenwood and Gleason 1955
Greenwood and Gleason 1955
Graver and Yackel 1968
Kalbfleisch 1966
McKay and Min 1992
Grinstead and Roberts 1982
Exoo 1989c, Radziszowski and Kreher 1988
Radziszowski and Kreher 1988
Exoo 1993, Radziszowski and Kreher 1988, Exoo 1998, Lesser 2001
Piwakowski 1996, Radziszowski and Kreher 1988
Exoo (unpub.), Radziszowski and Kreher 1988
Wang and Wang 1989, Radziszowski (unpub.), Lesser 2001
Wang and Wang 1989
Wang et al. 1994
Wang et al. 1994
Wang et al. 1994

20. Continuation...
p
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
q
20
21
22
23
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
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R(p, q)
[109, 209]
[122, 230]
[125, 252]
[136, 275]
18
25
[35, 41]
[49, 61]
[56, 84]
[73, 115]
[92, 149]
[97, 191]
[128, 238]
[133, 291]
[141, 349]
[153, 417]
[153, 815]
[182, 968]
[182, 1139]
Reference
Wang et al. 1994
Wang et al. 1994
Wang et al. 1994
Wang et al. 1994
Greenwood and Gleason 1955
McKay and Radziszowski 1995
Exoo (unpub.), McKay and Radziszowski 1995
Exoo 1989a, Mackey 1994
Exoo 1998, Exoo 2002
Radziszowski 1988, Mackey 1994
Piwakowski 1996, Mackey 1994, Harboth and Krause 2003
Piwakowski 1996, Spencer 1994, Burr et al. 1989
Su et al. 1998, Spencer 1994
Xu and Xie 2002
Xu and Xie 2002
Xu and Xie 2002
Luo et al. 2001

21. p
8
8
8
8
8
8
9
9
10
11
12
13
14
15
16
17
18
19
q
16
17
18
19
20
21
9
10
10
11
12
13
14
15
16
17
18
19
R(p, q)
[861, 170543]
[861, 245156]
[871, 346103]
[1054, 480699]
[1094, 657799]
[1328, 888029]
[565, 6588]
[580, 12677]
[798, 23556]
[1597, 184755]
[1637, 705431]
[2557, 2704155]
[2989, 10400599]
[5485, 40116599]
[5605, 155117519]
[8917, 601080389]
[11005, 2333606219]
[17885, 9075135299]
Reference
Xu and Xie 2002
Xu and Xie 2002
Xu and Xie 2002
Su et al. 2002
Su et al. 2002
Shearer 1986, Shi and Zheng 2001
Xu and Xie 2002
Shearer 1986, Shi 2002
Mathon 1987
Xu and Xie 2002
Mathon 1987
Mathon 1987
Mathon 1987
Mathon 1987
Luo et al. 2002
Luo et al. 2002
Luo et al. 2002
more at http://mathworld.wolfram.com/RamseyNumber.html
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22. References
• Jaam, M. J. (2006). A new construction technique of a triangle-free
3-colored K16s. Qatar. Elsevier
• Leader, Imre (2001). Friends and Strangers. Plus Magazine.
http://plus.maths.org/content/friends-and-strangers
• http://cstheory.stackexchange.com/questions/
9500/application-of-ramsey-numbers
• http://en.wikipedia.org/wiki/Ramsey%27s_theorem
(Proof of Infinite Ramsey Theorem)
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23. Special Thanks
Thanks to these guys for helping me in my LateX graphs:
• Claudio Fiandrino
• Jake
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24. Exercises
Problems 34 and 35.
• R(3, 5) = 14
• R(4, 4) ≤ 18
• R(3, 6) ≤ 19
Answer
• Prove that R(3, 5) = 14
Proof. Using the theorem 3.5.1 we have.
R(3, 5) ≤ R(2, 5) + R(3, 4) = 5 + 9
R(3, 4) was already shown in page 135.
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25. Thank you for Listening!
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