Papers › Upper bounds on chromatic number of 𝔼ⁿ in low dimensions
Upper bounds on chromatic number of 𝔼ⁿ in low dimensions
Andrii Arman, Andriy V. Bondarenko, Andriy Prymak, Danylo Radchenko
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Let χ(𝔼ⁿ) denote the chromatic number of the Euclidean space 𝔼ⁿ, i.e., the smallest number of colors that can be used to color 𝔼ⁿ so that no two points unit distance apart are of the same color. We present explicit constructions of colorings of 𝔼ⁿ based on sublattice coloring schemes that establish the following new bounds: χ(𝔼⁵)≤140, χ(𝔼ⁿ)≤7^(n/2) for n∈{6,8,24}, χ(𝔼⁷)≤1372, χ(𝔼⁹)≤17253, and χ(𝔼ⁿ)≤3ⁿ for all n≤38 and n=48,49.
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