  Let $P$ be a set of $n\geq 4$ points in general position in the plane.
Consider all the closed straight line segments with both endpoints in $P$.
Suppose that these segments are colored with the rule that disjoint segments
receive different colors. In this paper we show that if $P$ is the point
configuration known as the double chain, with $k$ points in the upper convex
chain and $l \ge k$ points in the lower convex chain, then $k+l- \left\lfloor
\sqrt{2l+\frac{1}{4}} - \frac{1}{2}\right\rfloor$ colors are needed and that
this number is sufficient.
