Answering a question posed by Bergelson and Leibman in , we establish a nilpotent version of the Polynomial Hales–Jewett Theorem that contains the main theorem in  as a special case. Important to the formulation and the proof of our main theorem is the notion of a relative syndetic set (relative with respect to a closed non-empty subsets of βG) . As a corollary of our main theorem we prove an extension of the restricted van der Waerden Theorem to nilpotent groups, which involves nilprogressions.
- Algebra in the Stone–Čech compactification
- Nilpotent groups
- Polynomial Hales–Jewett Theorem
- Ramsey theory
- Syndetic sets
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