Notre Dame Journal of Formal Logic

Proof Mining in Topological Dynamics

Philipp Gerhardy

Abstract

A famous theorem by van der Waerden states the following: Given any finite coloring of the integers, one color contains arbitrarily long arithmetic progressions. Equivalently, for every q,k, there is an N = N(q,k) such that for every q-coloring of an interval of length N one color contains a progression of length k. An obvious question is what is the growth rate of N = 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's 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.

Article information

Source
Notre Dame J. Formal Logic, Volume 49, Number 4 (2008), 431-446.

Dates
First available in Project Euclid: 17 October 2008

https://projecteuclid.org/euclid.ndjfl/1224257541

Digital Object Identifier
doi:10.1215/00294527-2008-022

Mathematical Reviews number (MathSciNet)
MR2456658

Zentralblatt MATH identifier
1188.03043

Subjects
Primary: 03 03F

Citation

Gerhardy, Philipp. Proof Mining in Topological Dynamics. Notre Dame J. Formal Logic 49 (2008), no. 4, 431--446. doi:10.1215/00294527-2008-022. https://projecteuclid.org/euclid.ndjfl/1224257541

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