The van der Waerden theorem states that given any natural numbers $k$ and $r$, there exists a natural number $W=W(k,r)$ such that if the set $\{1,2\cdots W\}$ is divided into $r$ classes (also called colors) then there exists a $k$-term arithmetic progression in one class.
I am trying to prove the theorem according to the outline given here.
I have understood the proofs for $W(3,2) \le 325$ and $W(3, 3) \le 7(2·3^7+1)(2·3^{7·(2·3^7+1)}+1)$ and on similar lines have worked out a proof for $W(3,4)\le 9(2.4^9+1)(2.4^{9(2.4^9+1)}+1)(2.4^{9(2.4^9+1)(2.4^{9(2.4^9+1)}+1)}+1)$.
What I want to do is to prove the theorem in general on similar lines. The main problem is for me as to what will be the upper bound to start off with and then how to deal with the cumbersome notation.
I understand asking someone to prove the full result is not appropriate here, but if you can give me any help I will greatly appreciate it.
Thanks.