On Classical Ramsey Numbers
Abstract
This paper has two parts. In the first part the question, “Why is it impossible to color the edges of kr(p,q) without forming either a red kp or a blue kq ?” is answered while in the second the question, “What is the smallest value of n for which kn[pic]kp,kq?” is changed to equivalent forms.
Introduction
A graph G is composed of a finite set V of elements called vertices and a set E of lines joining pairs of distinct vertices called edges. We denote the graph whose vertex set is V and whose edge set is E by G= (V,E). By a kn we mean a graph with n vertices and all edges joining these vertices with each other.
eg.
. k1 k2 k3 k4
Fig.1
Ramsey’s theorem states that if p,q≥2 are integers, then there is a positive integer n such that if we color the edges of kn using two colors, red and blue, it is impossible to color the edges of the kn without forming either a red kp or a blue kq. In short we formulate it as kn[pic]kp,kq (read as kn arrows kp,kq). The smallest value of such n is denoted by r(p,q),known as the Ramsey number. A famous example for the two color Ramsey theorem is k6 which arrows k3,k3. To prove k6 [pic]k3,k3, let’s put 6 points on a plane and call one of them v. There are 5 edges joining v to the remaining 5 points. Let’s color them red or blue. At least 3 of them will have the same color, red, say. Consider the 3 vertices at the other ends of these 3 red edges:
v
Blue
Red Fig.2
If any of the edges joining these 3 vertices with each other is red, then we have a red triangle. On the other hand if there is no red edge, we get a blue triangle.
This paper has two parts. In the first part the question, “Why is it impossible to color the edges of kr(p,q) without forming either a red kp or a blue kq ?” is answered while in the second the question, “What is the smallest value of n for which kn[pic]kp,kq?” is changed to equivalent forms.
Introduction
A graph G is composed of a finite set V of elements called vertices and a set E of lines joining pairs of distinct vertices called edges. We denote the graph whose vertex set is V and whose edge set is E by G= (V,E). By a kn we mean a graph with n vertices and all edges joining these vertices with each other.
eg.
. k1 k2 k3 k4
Fig.1
Ramsey’s theorem states that if p,q≥2 are integers, then there is a positive integer n such that if we color the edges of kn using two colors, red and blue, it is impossible to color the edges of the kn without forming either a red kp or a blue kq. In short we formulate it as kn[pic]kp,kq (read as kn arrows kp,kq). The smallest value of such n is denoted by r(p,q),known as the Ramsey number. A famous example for the two color Ramsey theorem is k6 which arrows k3,k3. To prove k6 [pic]k3,k3, let’s put 6 points on a plane and call one of them v. There are 5 edges joining v to the remaining 5 points. Let’s color them red or blue. At least 3 of them will have the same color, red, say. Consider the 3 vertices at the other ends of these 3 red edges:
v
Blue
Red Fig.2
If any of the edges joining these 3 vertices with each other is red, then we have a red triangle. On the other hand if there is no red edge, we get a blue triangle.
References: 1. Richard A.Brualdi, Introductory Combinatorics, Prentice-Hall, 2nd edition, 1992. 2. Victor Bryant, Aspects of Combinatorics, Cambridge University Press, 1992. 3. Miklos Bona,A walk through Combinatorics, World Scientific Printers, 2002. 4. Chen Chuan-Chong and Koh Khee-Meng, Principles and techniques in Combinatorics, Continental Press Pte Ltd, 1992. 5. Daniel I.A. Cohen, Basic Techniques of Combinatorial Theory, John Wiley & Sons Inc. 6. Van Lint &Wilson, A Course in Combinatorics, Cambridge University Press, 1992.