Does Lance dislike Ramsey Theory Because he's colorblind?

BILL: Lance, my wife asked if you dislike Ramsey Theory because you are colorblind.

LANCE: (laughs) It's why I don't like Ramsey Theory talks--impossible for me to follow. But I don't actually dislike Ramsey theory. I just don't like it as much as you do.

BILL: Nobody does, except possibly Graham, Rothchild,  Spencer, and your average Hungarian first grader.

LANCE: To me Ramsey Theory had one useful cool idea: The Probabilistic Method, that made people actually think Ramsey Theory was far more interesting than it really is.

BILL: Um, that is bad history and bad mathematics.

LANCE: I learned Ramsey Theory from Spencer and his book entitled Ten Lectures on the Probabilistic Method. But the Probabilistic Method was far more interesting than the Ramsey Theory. I suspect this is common: learn Ramsey Theory because of the Probabilistic Method. And some people get suckered into thinking that Ramsey Theory is interesting or important or both. My favorite application of the Probabilistic Method has nothing to do with Ramsey theory: Lautemann's proof that \(BPP \subseteq \Sigma_2\).

BILL:  A few points to make here

a) Some people call the Prob Method an application of  Ramsey Theory. I do not. The Prob Method was developed by Erdos to get lower bounds on asymptotic Ramsey numbers and was then used for many other things, that you and others find far more interesting. That's great, but that's not really an application of Ramsey Theory.

b) If the prob method was not discovered by Erdos for Ramsey Theory, would it have been discovered later when it was needed for something you care about more? Probably. But it may have been much later.

c) Ramsey Theory has real applications. I have a website of them here. And there are more. For example---

LANCE: Bill, your graduate course in Ramsey theory is titled Ramsey Theory and its ``Applications''. So you do not believe there are real applications.

BILL: Depends how you define real and applications. I put it in quotes since many of the applications are to pure math. Some are to lower bounds in models of computation, but some may still consider that to not be a real application. Rather than debate with the students what a real application is, I put it in quotes.

LANCE: Are there any real applications? That is, applications that are not to pure math, and not to lower bounds.

BILL: I would think you would count lower bounds to be a real application. 

LANCE: I am asking on behalf of the unfortunate Programming Language Student who takes your class thinking there will be real applications- perhaps they missed the quotes around applications.

BILL: I know of one real application. And its to Programming Languages! Ramsey Theory has been used  to prove programs terminate. I wrote a survey of that line of research  here.

LANCE: One? There is only One real application? Besides the application of learning the probabilistic method so they can use the method for more interesting problems. Or to save the earth from aliens.

BILL: Judging a field of Pure Math by how many applications it has does not really make sense. I find the questions and answers themselves interesting. Here is a list of theorems in Ramsey Theory that I find interesting. Do you? (This might not be a fair question either since many theorems are interesting because of their proofs.) 

a) (Ramsey's Theorem, Infinite Case) For every 2-coloring of \(K_\omega\) there exists \(H\subseteq N\) such that \(H\) is infinite and every edge between vertices in \(H\) is the same color. My slides here

b) (Van Der Warden's Theorem) For every \(c,k\) there exists W such that for all c-coloring of  \(\{1,\ldots,W\} \) there exists \(a,d\), both \(\ge 1  \) such that

\(a, a+d, \ldots, a+(k-1)d\) are all the same color.  My slides here.

c) (Subcase of Poly VDW Thm) For every \(c\) there exists W such that for all c-colorings of \(\{1,\ldots,W)\} there exists \(a,d\), both  \(\ge 1\) such that 

\(a, a+d, a+d^2,\ldots,a+d^{100}\) are the same color. My slides here.

d) For all finite colorings of the Eucidean plane there exists three points, the same color, such that the area of the triangle formed is 1. My slides: here.

So Lance, do these Theorems interest you?

LANCE: Sorry I fell asleep after you said "Ramsey". Let me look. Your slides are terrible. All of the colors are the same!

BILL: Maybe my wife was right.