Two-colorings of normed spaces without long monochromatic unit arithmetic progressions

TL;DR

This paper constructs a two-coloring of high-dimensional Euclidean spaces that prevents long monochromatic unit arithmetic progressions, revealing new insights into coloring problems in normed spaces.

Contribution

It introduces a novel coloring method in normed spaces that avoids long monochromatic unit arithmetic progressions, extending previous combinatorial results.

Findings

Existence of two-colorings avoiding long monochromatic progressions

Bound on the diameter of sets with no monochromatic isometric copies

Implication for coloring all normed spaces to prevent long monochromatic progressions

Abstract

Given a natural $n$, we construct a two-coloring of $\mathbb{R}^n$ with the maximum metric satisfying the following. For any finite set of reals $S$ with diameter greater than $5^{n}$ such that the distance between any two consecutive points of $S$ does not exceed one, no isometric copy of $S$ 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.