A variant of the density Hales-Jewett theorem


In a recent paper 'A variant of the Hales-Jewett theorem', M. Beiglböck provides a version of the classic coloring result in which an instance of the variable in a word giving rise to a monochromatic combinatorial line can be moved around in a finite structure of specified type (for example, an arithmetic progression). We prove a density version of this result in which all instances of the variable may move. As an application, we obtain an elementary proof of a result of V. Bergelson stating that multiplicatively large subsets of N contain arbitrarily long geoarithmetic progressions. © 2010 London Mathematical Society.

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Bulletin of the London Mathematical Society