Restricted van der Waerden theorem for nilprogressions

TL;DR

This paper proves a conjecture that the restricted van der Waerden theorem holds for nilprogressions of any rank, extending previous results limited to rank 2.

Contribution

It confirms the conjecture that the theorem applies to all ranks, broadening the scope of nilprogression results in combinatorics.

Findings

Proves the van der Waerden theorem for nilprogressions of arbitrary rank.

Extends the nilpotent polynomial Hales-Jewett theorem to higher ranks.

Confirms the conjecture posed by Johnson and Richter.

Abstract

In [Adv. Math., 321 (2017) 269-286], using the theory of ultrafilters, J. H. Johnson Jr., and F. K. Richter proved the nilpotent polynomial Hales-Jewett theorem. Using this result they proved the restricted version of the van der Waerden theorem for nilprogressions of rank $2$ and conjectured that this result must hold for arbitrary rank. In this article, we give an affirmative answer to their conjecture.