Ramsey Theory

Consider this problem:

Let's say that there are some people in a room for which some of them know each other and others do not. How many people must there be to make sure that either 3 of them are strangers to each other, or 3 have all met each other?

Another way to consider this problem is:

There are some points in a space, each connected by line segments to all of the other points. If each segment is either blue or red, then how many points must there be to ensure that there is one full triangle (with vertices on the points) outlined by all blue or all red?

In theory, these problems are exactly the same. The points are obviously people, and the color of the lines represents whether they know each other. If one color forms a triangle, three people either know each other or are total strangers to one another. Simple, right?

Anyway, the answer is 6 people. This is called the Ramsey number for two-colored triangles, written as:
R(3,3)=6

Let's call a complete graph (where all points are connected) with n points Kn. For Ramsey notation, the number of values in the parentheses represent the number of colors; the values themselves signify the graphs that we're trying to force. So R(3,6)=18 would be two colored, either forcing a red triangle or a blue K6.

Here are the Ramsey numbers that we know for now:
3, 3 = 6
3, 3, 3 = 17
3, 4 = 9
3, 5 = 14
4, 4 = 18
3, 6 = 18
3, 7 = 23
3, 8 = 28
3, 9 = 36
4, 5 = 25
6, 7 = 298

But then there are the irregular graphs, like straight lines and bowties and things like that. Well, thank you, Stefan Burr!