Disjointness graphs of short polygonal chains

The {\em 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 {\em $\chi$-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) \emph{$2$-way infinite} polygonal chains of length $3$ is $\chi$-bounded: for every such graph $G$, we have $\chi(G)\le(\omega(G))^3+\omega(G).$