A famous theorem by van der Waerden states the following: Given any finite colouring of the integers, one colour contains arbitrar- ily long arithmetic progressions. Equivalently, for every q, k there is an N = N(q, k) such that for every q-colouring of an interval of length N one colour contains a progression of length k. An obvious question is: What is the growth rate of N(q, k)? Some proofs,
... [Show full abstract] like van der Waerden's combinatorial argument, answer this question directly, while the topo- logical proof by Furstenberg and Weiss does not. We present an analysis of (Girard's variant of) Furstenberg and Weiss' proof based on monotone functional interpretation, both yielding bounds and providing a general illustration of proof mining in topological dynamics. The bounds do not improve previous results by Girard, but only - as is also revealed by the analysis - because the combinatorial proof and the topological dynamics proof in principle are identical. "Proof mining" is the activity of extracting additional information from proofs in mathematics and computer science. The two main types of additional infor- mation that may be extracted are quantitative and qualitative information. An example of the former is extracting a rate of convergence from a proof that a certain iteration sequence in a compact metric space converges. An example of the latter is establishing that the convergence is uniform in the starting point of the iteration or that the result not only holds for compact metric spaces, but already for bounded ones. Naturally, when the proof to be analysed is constructive, we expect quantitative information - realizers or bounds - to be present explicitly in the proof, and qualitative strengthenings of the theorem may follow trivially from the exact structure of the realizers.