  Let $\omega(G)$ and $\chi(G)$ denote the clique number and chromatic number
of a graph $G$, respectively. The {\em disjointness graph} of a family of
curves (continuous arcs in the plane) is the graph whose vertices correspond to
the curves and in which two vertices are joined by an edge if and only if the
corresponding curves are disjoint. A curve is called {\em $x$-monotone} if
every vertical line intersects it in at most one point. An $x$-monotone curve
is {\em grounded} if its left endpoint lies on the $y$-axis.
  We prove that if $G$ is the disjointness graph of a family of grounded
$x$-monotone curves such that $\omega(G)=k$, then $\chi(G)\leq \binom{k+1}{2}$.
If we only require that every curve is $x$-monotone and intersects the
$y$-axis, then we have $\chi(G)\leq \frac{k+1}{2}\binom{k+2}{3}$. Both of these
bounds are best possible. The construction showing the tightness of the last
result settles a 25 years old problem: it yields that there exist $K_k$-free
disjointness graphs of $x$-monotone curves such that any proper coloring of
them uses at least $\Omega(k^{4})$ colors. This matches the upper bound up to a
constant factor.
