Woodall  has shown that the rational points in the plane may be
colored with two colors so that any two points at distance 1 have
Johnson  extended this to show that there is a two-coloring that
"forbids" not only the distance 1 , but also every distance of the form
√(p/q) , where p and
q are odd positive integers.
Johnson's coloring also forbids some other distances, such as
√6 , because no two rational points are precisely
that far apart.
We show that Johnson's coloring is optimal, in the sense that, if
d actually occurs as a distance, and d is not of the form
√(p/q) , for odd positive integers
p and q , then no two-coloring of
Q² forbids both the distance 1 and d .
This settles one of Johnson's questions.
We also consider two-colorings of Q³ ,
and solve two more of his problems.