The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number $\chi (G)$ is upper bounded by a function of its clique number $\omega (G)$.

Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have $\chi (G)\le {(\omega (G))}^3+\omega (G)$.

集合系统的不相交图是这样的图，其顶点是集合，当且仅当它们不相交时，两个集合通过边连接。已知平面内任意线段系统的不相交图G都是χ 有界，即它的色数$\chi （G）$上限由其团数的函数决定$\omega （G）$。

在这里，我们证明这个说法对于长度为 2 的多边形链系统并不成立。我们还构造了长度为 3 的多边形链系统，使得它们的不相交图具有任意大的周长和色数。在相反的方向上，我们证明长度为 3 的（可能自相交）2路无限多边形链的不相交图的类是χ有界的：对于每个这样的图G，我们有$\chi （G）\le {（\omega （G））}^3+\omega （G）$。