Chapter 33 Ramsey's Theory

 
       Chapter 33 is about Ramsey's theory. It is named after Frank Plumpton Ramsey who died at the age of 26. His theory is based on graph theory. The main idea of this is studying sets of points joined by lines. In this theory it compares points to represent people at a dinner table such as six points representing six people. It compares the points drawn to lines with two different colors. These represent people who know each other at the dinner table and people who are strangers.

        This chapter talks about how Kn would represent the graph being that n represents the number of points the graph has and k is just a variable to represent the graph. It then talks about coloring the lines red and blue to represent two different groups and that colored with three would represent three different groups. It also talks about subsets and how each line connected to two points represents a subset. To further that a subgraph is a graph contained in the graph.

            Ramsey’s theory can be compared to a game. The most common is called Sim it is played on K6 ( a graph with 6 points). Which can be compared to the problem with the six people. It also can be put in R(3,3)=6 notation. Which the R represents Ramsey’s number, the three represents a triangle of one graph or a subgraph, and the other three represents a triangle of another color. The next thing talks about Knuth’s arrow notation which is 3 arrow 3 is the same thing as 3*3*3=27. Also 3 arrow arrow 3 is the same as 3 arrow 27 or 3^27 which is the same as 3^3^3. This number written as a number is 7,625,597,484,987.

            Ramsey’s theory poses a question. An example of a question it poses is while having a party and you want to invite at least four people who know each other and five who don’t, how many people should you invite? The answer is 25 people. The way it is answered is by drawing points where each point represents a person and each point is joined to all of the other points by a line. An example of this is if you color a line red it means two points represent two people who know each other. If the line is blue it means that the two people do not know each other. The way 25 is proved as the answer is because of asking the question what the smallest amount of points are needed so that there are four points joined by red lines and five joined by blue lines? This was not proved until recently. This can also be written in the R(4,5)=25 notation. Where the R represents Ramsey’s number, the four represents one subgroup, and the five represents another subgroup.

            In general there were some confusing ideas presented in this chapter and it was interesting that the graph could be compared to a real world situation. Such as inviting people to a party and how many to invite if you want a certain amount of people to know each other and a certain amount to not know each other. It was surprising in a way. Questions the chapter posed for me were things like knowing when a graph is six colored and weather you are certain to get a subgraph of one of the colors.