What is Combinatorics?

So the title "Combinatorial Hijinks" comes from the fact that Combinatorics is the field of mathematics that I study (and the fact that Hijinks is a cool word). You might ask, what is combinatorics? Well Wikipedia has this to say:

"Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects."

By the way, isn't wikipedia neat? Who would have thought that an encyclopedia edited by anyone would be so useful...

Ok, my tangent is over (though my secant has just begun; expect other math jokes to work their way into my posts). I would describe combinatorics as the art of counting without actually counting, of determining structure and properties of objects without actually knowing very much about those objects. An example I like to give comes from Ramsey Theory.

The idea is this: you are throwing a party, and want to know the least number of people you need to invite so that their is a group of three people who all know each other, or a group of three people such that none of them know each other. Why you would want to do such a thing is not important, using such language as "party" helps make the problem seem more "applied"...

Anyhoo, it turns out six people is enough; let's see why. Let's call the six people Alice, Bob, Christine, David, Edward and Frank. As mathematicians we love to do this sort of thing, so we can shorten the names to A, B, C, D, E and F. Consider Alice. Of the other five people, she either knows at least three of them, or there are three she doesn't know. Without loss of generality (a favorite phrase amongst mathematicians) consider the case where she knows Bob, Christine and David. Now, suppose Bob and Christine also know each other, then Alice, Bob and Christine form a group of three people who all know each other. This will work if any two of Bob, Christine and David know each other. So the only other possibility is that the three of them are mutual strangers, in which case Bob, Christine and David form a group of three people who each don't know each other. Voila! Note that the case where Alice knows two or less people is analogous, since she would have to not know 3 people, and we could repeat the above proof, switching the term "know" with "doesn't know." That is what is meant by "without loss of generality."

The cool thing about that problem is that we don't have to know anything about the structure of these people's lives to conclude something definite about them. The general theme of Ramsey theory is that if you enough objects, there will be order. And that is pretty cool, that we can know things even when we don't know anything.