Two-colorings of normed spaces without long monochromatic unit arithmetic progressions
arXiv:2203.04555 · doi:10.1137/22M1483700
Abstract
Given a natural , we construct a two-coloring of with the maximum metric satisfying the following. For any finite set of reals with diameter greater than such that the distance between any two consecutive points of does not exceed one, no isometric copy of is monochromatic. As a corollary, we prove that any normed space can be two-colored such that all sufficiently long unit arithmetic progressions contain points of both colors.
15 pages. v2: a few modifications based on the reviews