Geodesic
On the chromatic number of slices without monochromatic unit arithmetic progressions
Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 78-84

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For $h,n\geq 1$ and $e>0$ we consider a chromatic number of the spaces $\mathbb{R}^n\times[0, e]^h$ and general results in this problem. Also we consider the chromatic number of normed spaces with forbidden monochromatic arithmetic progressions. We show that for any $n$ there exists a two-coloring of $\mathbb{R}^n$ such that all long unit arithmetic progressions contain points of both colors and this coloring covers spaces of the form $\mathbb{R}^n\times[0, e]^h$.
Keywords: chromatic number, Hadwiger–Nelson problem. 
@article{CHEB_2023_24_4_a6,
     author = {V. O. Kirova},
     title = {On the chromatic number of slices without monochromatic unit arithmetic progressions},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {78--84},
     publisher = {mathdoc},
     volume = {24},
     number = {4},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a6/}
}
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V. O. Kirova. On the chromatic number of slices without monochromatic unit arithmetic progressions. Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 78-84. http://geodesic.mathdoc.fr/item/CHEB_2023_24_4_a6/