Geodesic
On the chromatic numbers of integer and rational lattices
Contemporary Mathematics. Fundamental Directions, Topology, Tome 51 (2013), pp. 110-122.

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In this paper, we give new upper bounds for the chromatic numbers for integer lattices and some rational spaces and other lattices. In particular, we have proved that for any concrete integer number $d$, the chromatic number of $\mathbb Z^n$ with critical distance $\sqrt{2d}$ has a polynomial growth in $n$ with exponent less than or equal to $d$ (sometimes this estimate is sharp). The same statement is true not only in the Euclidean norm, but also in any $l_p$ norm. Moreover, we have given concrete estimates for some small dimensions as well as upper bounds for the chromatic number of $\mathbb Q_p^n$, where by $\mathbb Q_p$ we mean the ring of all rational numbers having denominators not divisible by some prime numbers. 
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V. O. Manturov. On the chromatic numbers of integer and rational lattices. Contemporary Mathematics. Fundamental Directions, Topology, Tome 51 (2013), pp. 110-122. http://geodesic.mathdoc.fr/item/CMFD_2013_51_a6/

[1] Kupavskii A. B., “O raskraskakh sfer, vlozhennykh v $\mathbb R^n$”, Mat. sb., 202:6 (2011), 83–110 | DOI | MR | Zbl

[2] Raigorodskii A. M., “Problema Borsuka i khromaticheskie chisla nekotorykh metricheskikh prostranstv”, Usp. mat. nauk, 56:1(337) (2001), 107–146 | DOI | MR | Zbl

[3] Raigorodskii A. M., “O khromaticheskom chisle prostranstva s $l_p$-normoi”, Usp. mat. nauk, 59:5 (2004), 161–162 | DOI | MR | Zbl

[4] Raigorodskii A. M., Lineino-algebraicheskie metody v kombinatorike, MTsNMO, M., 2007 | Zbl

[5] Raiskii D. E., “Realizatsiya vsekh rasstoyanii pri razbienii prostranstva $R^n$ na $n+1$ chast”, Mat. zametki, 7:3 (1970), 319–323 | MR | Zbl

[6] Benda M., Perles M., “Introduction to colorings of metric spaces”, Geombinatorics, 9 (2000), 111–126 | MR

[7] Brass P., Moser L., Pach J., Research Problems in Discrete Geometry, Springer, 2005 | MR

[8] Cibulka J., “On the chromatic numbers of real and rational spaces”, Geombinatorics, 18 (2008), 53–65 | MR | Zbl

[9] De Bruijn N. G., Erdős P., “A colour problem for infinite graphs and a problem in the theory of relations”, Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951), 371–373 | Zbl

[10] Frankl P., Wilson R. M., “Intersection theorems with geometric consequences”, Combinatorica, 1 (1981), 357–368 | DOI | MR | Zbl

[11] Fűredi Z., Kang J.-H., “Distance graphs on $\mathbb Z^n$ with $l_1$-norm”, Theor. Computer Sci., 319 (2004), 357–366 | DOI | MR

[12] Larman D. G., Rogers C. A., “The realization of distances within sets in euclidean spaces”, Mathematica, 19 (1972), 1–24 | MR | Zbl

[13] Moser L., Moser W., “Solution to Problem 10”, Canad. Math. Bull., 4 (1961), 187–189

[14] O'Bryant K., A complete annotated bibliography of work related to Sidon sequences, 2011, arXiv: math/0407117[math.NT]

[15] Ruzsa I., “Solving a linear equation in a set of integers. II”, Acta Arith., 65:3 (1993), 259–282 | MR | Zbl

[16] Ruzsa I., “Solving a linear equation in a set of integers. II”, Acta Arith., 72:4 (1995), 385–397 | MR | Zbl