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 CMFD, 2013, Volume 51, Pages 110–122 (Mi cmfd257)

On the chromatic numbers of integer and rational lattices

V. O. Manturov

Faculty of Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Moscow, Russia

Abstract: 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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English version:
Journal of Mathematical Sciences, 2016, 214:5, 687–698

UDC: 519.1

Citation: V. O. Manturov, “On the chromatic numbers of integer and rational lattices”, Topology, CMFD, 51, PFUR, M., 2013, 110–122; Journal of Mathematical Sciences, 214:5 (2016), 687–698

Citation in format AMSBIB
\Bibitem{Man13}
\by V.~O.~Manturov
\paper On the chromatic numbers of integer and rational lattices
\inbook Topology
\serial CMFD
\yr 2013
\vol 51
\pages 110--122
\publ PFUR
\mathnet{http://mi.mathnet.ru/cmfd257}
\transl
\jour Journal of Mathematical Sciences
\yr 2016
\vol 214
\issue 5
\pages 687--698
\crossref{https://doi.org/10.1007/s10958-016-2806-1}