V. A. Voronov, A. Ya. Kanel-Belov, G. A. Strukov, D. D. Cherkashin, “On the chromatic numbers of <nobr>$3$</nobr>-dimensional slices”, Zap. Nauchn. Sem. POMI, 2022, Volume 518,Pages <nobr>94

On the chromatic numbers of $3$-dimensional slices

V. A. Voronov^ab, A. Ya. Kanel-Belov^b, G. A. Strukov^c, D. D. Cherkashin^d

^a Caucasus Mathematical Center, Adyghe State University, Maikop
^b Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region
^c St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
^d Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Abstract: We prove that for an arbitrary $\varepsilon > 0$ holds
$$ \chi (\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10, $$
where $\chi(M)$ stands for the chromatic number of an (infinite) graph with the vertex set $M$ and the edge set consists of pairs of points at the distance $1$ apart.

Key words and phrases: distance graphs, chromatic number of space.

UDC: 514.17, 519.174, 515.124.3

Received: 01.12.2022