Generalization of Ramsey NumbersFurther Research on Ramsey Theorem

Asaad, Al-Ahmadgaid B.

email: alstated@gmail.com

Mindanao State UniversityIligan Institute of Technology

Generalization

The definition of the Ramsey number R(p, q) with 2 parameters can begeneralized in a natural way to the Ramsey number R(p1, p2, · · · , pk)with k parameters as follows. Let k , p1, p2, · · · , pk ε N with k ≥ 3.The Ramsey number R(p1, p2, · · · , pk) is the smallest natural numbern 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 thepi -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 thatR(3, 3, 3) ≥ 17.

• Thus, R(3, 3, 3) = 17

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Generalization

The definition of the Ramsey number R(p, q) with 2 parameters can begeneralized in a natural way to the Ramsey number R(p1, p2, · · · , pk)with k parameters as follows. Let k , p1, p2, · · · , pk ε N with k ≥ 3.The Ramsey number R(p1, p2, · · · , pk) is the smallest natural numbern 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 thepi -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 thatR(3, 3, 3) ≥ 17.

• Thus, R(3, 3, 3) = 17

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Generalization

The definition of the Ramsey number R(p, q) with 2 parameters can begeneralized in a natural way to the Ramsey number R(p1, p2, · · · , pk)with k parameters as follows. Let k , p1, p2, · · · , pk ε N with k ≥ 3.The Ramsey number R(p1, p2, · · · , pk) is the smallest natural numbern 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 thepi -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 thatR(3, 3, 3) ≥ 17.

• Thus, R(3, 3, 3) = 17

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Example

From Exercise 42. Show that R(3, 3, 2) = 6

Proof. Let v0 be a vertex from any of the six vertices. Joining thevertex 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 witheither of the three colours.

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Assuming the edges were {(v0,v1), (v0,v2), and (v0,v3)} and that it iscoloured with red. Then

v0

v1 v2 v3

If any of the edges between v1, v2, and v3 is coloured with red, thenwe can form a monochromatic triangle of colour red.

v0

v1 v2 v3

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If none of them is coloured with red, then either blue or yellow isthe colour of the edges. Suppose the colour of the edge between thevertices 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 edgecoloured with yellow.

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Suppose, the edges between the vertices v1, v2, and v3 are colouredwith blue. If the edge formed between vertices v1 and v3 is colouredwith red. Then, we have a single monochromatic triangle colouredwith red. But, if the edge is coloured with blue, then we can havea monochromatic triangle of colour blue. In addition, if the edge iscoloured with yellow, then we have a single line (edge) coloured withyellow.

v0

v1 v2 v3

v0

v1 v2 v3

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v0

v1 v2 v3

Hence, using 6 vertices is enough to show that there exist 3-clique ofblue, 3-clique of red, or 2-clique of yellow. �

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Applications of Ramsey Numbers

These are the areas where we can apply Ramsey numbers

• Number Theory

• Geometry

• Communications

• Decision Making

• Computer Sciencehttp://www.cs.umd.edu/~gasarch/ramsey/ramsey.html

• and more...

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Applications of Ramsey Numbers

These are the areas where we can apply Ramsey numbers

• Number Theory

• Geometry

• Communications

• Decision Making

• Computer Sciencehttp://www.cs.umd.edu/~gasarch/ramsey/ramsey.html

• and more...

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Applications of Ramsey Numbers

These are the areas where we can apply Ramsey numbers

• Number Theory

• Geometry

• Communications

• Decision Making

• Computer Sciencehttp://www.cs.umd.edu/~gasarch/ramsey/ramsey.html

• and more...

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Applications of Ramsey Numbers

These are the areas where we can apply Ramsey numbers

• Number Theory

• Geometry

• Communications

• Decision Making

• Computer Sciencehttp://www.cs.umd.edu/~gasarch/ramsey/ramsey.html

• and more...

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Applications of Ramsey Numbers

These are the areas where we can apply Ramsey numbers

• Number Theory

• Geometry

• Communications

• Decision Making

• Computer Sciencehttp://www.cs.umd.edu/~gasarch/ramsey/ramsey.html

• and more...

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Applications of Ramsey Numbers

These are the areas where we can apply Ramsey numbers

• Number Theory

• Geometry

• Communications

• Decision Making

• Computer Sciencehttp://www.cs.umd.edu/~gasarch/ramsey/ramsey.html

• and more...

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Ramsey Theorem

In combinatorics, Ramsey’s theorem states that in any colouring of theedges of a sufficiently large complete graph, one will find monochro-matic complete subgraphs. For two colours, Ramsey’s theorem statesthat for any pair of positive integers (p, q), there exists a least positiveinteger 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 completesubgraph on p vertices which is entirely blue, or a complete subgraphon q vertices which is entirely red.

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More on Theorem

• Extension of the TheoremThe theorem can also be extended to hypergraphs. An m-hypergraphis a graph whose ”edges” are sets of m vertices - in a normal graphan edge is a set of 2 vertices.

• Infinite Ramsey theoremA further result, also commonly called Ramsey’s theorem, appliesto infinite graphs. In a context where finite graphs are also beingdiscussed it is often called the ”Infinite Ramsey theorem”.

Theorem: Let X be some countably infinite set and colour theelements 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 nsubsets of M all have the same colour.

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List of Ramsey NumbersList of Ramsey numbers for p-clique and q-clique, p, q ≤ 19

p q R(p, q) Reference3 3 6 Greenwood and Gleason 19553 4 9 Greenwood and Gleason 19553 5 14 Greenwood and Gleason 19553 6 18 Graver and Yackel 19683 7 23 Kalbfleisch 19663 8 28 McKay and Min 19923 9 36 Grinstead and Roberts 19823 10 [40, 43] Exoo 1989c, Radziszowski and Kreher 19883 11 [46, 51] Radziszowski and Kreher 19883 12 [52, 59] Exoo 1993, Radziszowski and Kreher 1988, Exoo 1998, Lesser 20013 13 [59, 69] Piwakowski 1996, Radziszowski and Kreher 19883 14 [66, 78] Exoo (unpub.), Radziszowski and Kreher 19883 15 [73, 88] Wang and Wang 1989, Radziszowski (unpub.), Lesser 20013 16 [79, 135] Wang and Wang 19893 17 [92, 152] Wang et al. 19943 18 [98, 170] Wang et al. 19943 19 [106, 189] Wang et al. 1994

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Continuation...

p q R(p, q) Reference3 20 [109, 209] Wang et al. 19943 21 [122, 230] Wang et al. 19943 22 [125, 252] Wang et al. 19943 23 [136, 275] Wang et al. 19944 4 18 Greenwood and Gleason 19554 5 25 McKay and Radziszowski 19954 6 [35, 41] Exoo (unpub.), McKay and Radziszowski 19954 7 [49, 61] Exoo 1989a, Mackey 19944 8 [56, 84] Exoo 1998, Exoo 20024 9 [73, 115] Radziszowski 1988, Mackey 19944 10 [92, 149] Piwakowski 1996, Mackey 1994, Harboth and Krause 20034 11 [97, 191] Piwakowski 1996, Spencer 1994, Burr et al. 19894 12 [128, 238] Su et al. 1998, Spencer 19944 13 [133, 291] Xu and Xie 20024 14 [141, 349] Xu and Xie 20024 15 [153, 417] Xu and Xie 20024 16 [153, 815]4 17 [182, 968] Luo et al. 20014 18 [182, 1139]

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p q R(p, q) Reference8 16 [861, 170543]8 17 [861, 245156] Xu and Xie 20028 18 [871, 346103] Xu and Xie 20028 19 [1054, 480699] Xu and Xie 20028 20 [1094, 657799] Su et al. 20028 21 [1328, 888029] Su et al. 20029 9 [565, 6588] Shearer 1986, Shi and Zheng 20019 10 [580, 12677] Xu and Xie 2002

10 10 [798, 23556] Shearer 1986, Shi 200211 11 [1597, 184755] Mathon 198712 12 [1637, 705431] Xu and Xie 200213 13 [2557, 2704155] Mathon 198714 14 [2989, 10400599] Mathon 198715 15 [5485, 40116599] Mathon 198716 16 [5605, 155117519] Mathon 198717 17 [8917, 601080389] Luo et al. 200218 18 [11005, 2333606219] Luo et al. 200219 19 [17885, 9075135299] Luo et al. 2002

more at http://mathworld.wolfram.com/RamseyNumber.html

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References

• Jaam, M. J. (2006). A new construction technique of a triangle-free3-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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Special Thanks

Thanks to these guys for helping me in my LateX graphs:

• Claudio Fiandrino

• Jake

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Exercises

Problems 34 and 35.

• R(3, 5) = 14

• R(4, 4) ≤ 18

• R(3, 6) ≤ 19

Answer

• Prove that R(3, 5) = 14Proof. 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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Thank you for Listening!

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