Let $\omega (G)$ and $\chi (G)$ denote the clique number and chromatic number of a graph G, respectively. The 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 x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is 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 $\chi (G)\le \left(\begin{array}{c}k+1\\ 2\end{array}\right)$. If we only require that every curve is x-monotone and intersects the y-axis, then we have $\chi (G)\le \frac{k+1}{2}\left(\begin{array}{c}k+2\\ 3\end{array}\right)$. 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 $\mathrm{\Omega }(k^4)$ colors. This matches the upper bound up to a constant factor.

让 $\omega （G）$ 和 $\chi （G）$分别表示图G的集团数和色数。一组曲线的不相交图（平面中的连续弧）是这样的图，其顶点与曲线相对应，并且当且仅当相应的曲线不相交时，两个顶点通过一条边连接。如果一条曲线的每条垂直线最多在一个点处相交，则该曲线称为x单调。如果x单调曲线的左端点位于y轴，则该曲线接地。

我们证明如果G是接地的x-单调曲线族的不相交图，则$\omega （G）=ķ$， 然后 $\chi （G）\le （\begin{array}{c}ķ+\mathrm{1个}\\ 2\end{array}）$。如果只要求每条曲线是x-单调并且与y-轴相交，则我们有$\chi （G）\le \frac{ķ+\mathrm{1个}}{2}（\begin{array}{c}ķ+2\\ 3\end{array}）$。这两个界限都是最好的。显示最后结果紧密度的构造解决了一个25年的问题：它产生了存在${ķ}_{ķ}$的-free不相交曲线X -monotone曲线，使得它们中的任何适当的着色使用至少$\mathrm{\Omega }（{ķ}^4）$颜色。这使上限匹配一个恒定因子。