On Classical Ramsey Numbers

Only available on StudyMode
  • Topic: Color, Ramsey's theorem, Ramsey theory
  • Pages : 10 (2662 words )
  • Download(s) : 62
  • Published : March 25, 2013
Open Document
Text Preview

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.


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.


k1 k2
k3 k4


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:




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. A complete subgraph with all its edges having the same color is called a monochromatic sub graph or a monochromatic clique. The word coloring refers to assigning either of the colors red or blue for the edges and is used in a different sense from edge coloring; i.e. adjacent edges can have the same color. A kn with one side missing is one which becomes a kn when the missing side is filled. Let’s denote it by kn-s. Clearly a k1 & a k2 with one side missing are vacuous sets.

A k3 -s A k4-s A k5 -s Fig.3

The main result
We have shown that k6[pic]k3,k3. But why is it impossible to color the edges of k6 with out forming a monochromatic k3 of either color? In a k6 there are 6C3=20 triangles. If these triangles were drawn separately, they would have 20[pic]3=60 edges. But a k6 has only 6C2= 15 edges. Hence an edge is shared by [pic] = 4 triangles in a k6. Now let’s try to form a k6 whose edges colored with red and blue but doesn’t consist of a monochromatic k3 of either color. What problem shall we encounter? Let’s start with a red edge, v1v2. As shown below, it is shared by 4 triangles. Let all triangles on v1v2 have blue sides, except v1v2.

v1 v1

v3 v5 v3 v5

v4 v6 v4 v6

tracking img