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math.MG2022★ 6 cited
Coverings of planar and three-dimensional sets with subsets of smaller diameter
Alexander Tolmachev, Dmitry Protasov, Vsevolod Voronov
Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given there is a minim…
math.CO2022
On the chromatic numbers of 3-dimensional slices
D. D. Cherkashin, A. J. Kanel-Belov, G. A. Strukov +1
We prove that for an arbitrary holds \[ χ(\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10, \] where stands for the chromatic number of an (infinite) graph w…
math.CO2022
On the chromatic number of 2-dimensional spheres
Danila Cherkashin, Vsevolod Voronov
In 1976 Simmons conjectured that every coloring of a 2-dimensional sphere of radius strictly greater than in three colors has a couple of monochromatic points at the distance…