Coloring the Line
Kemnitz and Marangio showed that for any list of k distances, the real
line can be colored with k+1 colors so that all k of the distances are
"forbidden," meaning that two points any of those distances apart
must be colored differently. Their proof is non-constructive, as it appeals to
a famous theorem of de Bruijn and Erdos,
from the proof of which the axiom of choice cannot be excised. Here, we give
explicit instructions for obtaining distance-forbidding colorings. Also, we
examine extensions to periodic colorings.