Coloring Curves that Cross a Fixed Curve

We prove that for every integer t⩾ 1 , the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is χ-bounded. This is essentially the strongest χ-boundedness result one can get for those kind of graph classes. As a corollary, we prove that for any fixed integers k⩾ 2 and t⩾ 1 , every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has O(nlog n) edges.