 
Summary: ON THE UPPER CHROMATIC NUMBERS OF THE
REALS
AARON F. ARCHER
Abstract. Let S be a metric space and let k be a positive in
teger. Define “
Ø (k) (S) to be the smallest positive integer m such
that for every k \Theta m array D = (D ij ) of positive real numbers, S
can be colored with the colors 1; 2; : : : ; m such that no two points
of distance D ij are both colored j. We improve the best upper
bound known on “
Ø (k) (R) from 32 k k! to d4eke, where e is the base
of the natural logarithm. We prove a conjecture of A. Abrams
that “
Ø (k) (Z) = “
Ø (k) (R) for all k 2 N, extend this result to higher
dimensions under the l 1 and l 1 norms, and prove that the upper
chromatic numbers are finite for these spaces. We also introduce
a new related chromatic quantity of a graph G, the chromatic ca
pacity, Ø cap (G).
1. Introduction
