On the chromatic number and equilateral dimension of $\mathbb{R}^n$ with the tropical norm

Abstract

We study the tropical chromatic number of $\mathbb{R}^n$, $χ_{\mathrm{tr}}(\mathbb{R}^n)$, the tropical analogue of the well-known Hadwiger-Nelson problem in $\mathbb{R}^2$. An upper bound to $χ_{\mathrm{tr}}(\mathbb{R}^n)$ is $2^n$. It is conjectured that $χ_{\mathrm{tr}}(\mathbb{R}^n) = 2^n$, which is known to be the case for the measurable chromatic number. Asymptotically we get that $\displaystyle χ_{\mathrm{tr}}(\mathbb{R}^n) = Θ\!\left(\frac{2^n}{\sqrt{n}}\right)$. By constructing a graph with 62 vertices and 577 edges we demonstrate that $χ_{\mathrm{tr}}(\mathbb{R}^3)=8$. A related problem is the tropical equilateral dimension of $\mathbb{R}^n$, $\mathrm{e}_{\mathrm{tr}}(\mathbb{R}^n)$, the maximum size of a set $S$ of points of the same tropical distance from one another. We show that $\displaystyle \mathrm{e}_{\mathrm{tr}}(\mathbb{Z}^n) \geq \binom{n+1}{\lfloor (n+1)/2 \rfloor}$ and conjecture that $\mathrm{e}_{\mathrm{tr}}(\mathbb{R}^n)$ is exactly this Sperner's antichain bound. The conjecture is verified in dimension $n \leq 3$ and also when $S$ is an equilateral set of tropical distance $2R$ contained in a tropical sphere of radius $R$. We are not aware of the appearance of Sperner's bound in the context of chromatic number or equilateral set.