Erdős Problem #704

declared status 'open'. Formalized: no. Let $G_n$ be the unit distance graph in $\mathbb{R}^n$, with two vertices joined by an edge if and only if the distance between them is $1$. Estimate the chromatic number $\chi(G_n)$. Does it grow exponentially in $n$? Does$$\lim_{n\to \infty}\chi(G_n)^{1/n}$$exist? Current best: Prosanov [Pr20] has given an alternative proof of this upper bound. [Pr20] Prosanov, Roman, A new proof of the Larman-Rogers upper bound for the chromatic number of the Euclidean space. Prize: no. Tags: chromatic number, geometry, graph theory.

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