Coloring curves that cross a fixed curve

We prove that for every integer $t\geq 1$, 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 $\chi$-bounded. This is essentially the strongest $\chi$-boundedness result one can get for this kind of graph classes. As a corollary, we prove that for any fixed integers $k\geq 2$ and $t\geq 1$, every $k$-quasi-planar topological graph on $n$ vertices with any two edges crossing at most $t$ times has $O(n\log n)$ edges.Comment: Small corrections, improved presentatio

Coloring curves that cross a fixed curve

We prove that for every integer t\geqslant 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 $k\geqslant$ 2 and $t\geqslant$ 1, every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has $O(nlogn)$ edges

Note on the number of edges in families with linear union-complexity

We give a simple argument showing that the number of edges in the intersection graph $G$ of a family of $n$ sets in the plane with a linear union-complexity is $O(\omega(G)n)$. In particular, we prove $\chi(G)\leq \text{col}(G)< 19\omega(G)$ for intersection graph $G$ of a family of pseudo-discs, which improves a previous bound.Comment: background and related work is now more complete; presentation improve