Restricted van der Waerden theorem for nilprogressions
For k≥2k\geq 2k≥2, let ∑<k\sum_{<k}∑<k be the collection of words in ∗1,…,∗d*_1,\ldots,*_d∗1,…,∗d in which each letter occurs at most k−1k-1k−1 times. For elements a,x1,…,xda,x_1,\ldots,x_da,x1,…,xd of an sss-step nilpot…