On Chromatic Numbers of Integer and Rational Lattices

TL;DR

This paper establishes new upper bounds for the chromatic numbers of integer and rational lattices across various norms, demonstrating polynomial growth in dimension and providing specific estimates for small cases.

Contribution

It introduces novel upper bounds for lattice chromatic numbers, applicable to integer and rational lattices in multiple norms, with polynomial growth estimates and specific dimension bounds.

Findings

Chromatic number of Z^n grows polynomially with dimension, exponent .

Results hold in Euclidean and l_p norms.

Provided concrete bounds for small dimensions and rational lattices.

Abstract

In the present paper, we have found new upper bounds for chromatic numbers for integer lattices and some rational spaces and other lattices. In particular, we have proved that for any concrete critical distance $d$ the chromatic number of $\Z^{n}$ 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. Besides, we have given concrete estimates for some small dimensions as well as upper bounds for the chromatic number of $\Q_{p}^{n}$, where by $\Q_{p}$ we mean the ring of all rational numbers having denominators not divisible by some prime numbers.