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Proof Mining in Topological Dynamics

A famous theorem by van der Waerden states the following: Given any finite colouring of the integers, one colour contains arbitrarily 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, like van der Waerden’s combinatorial argument, answer this question directly, while the topological 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.