Coloring -free intersection graphs of geometric objects in the plane

The intersection graph of a collection C of sets is the graph on the vertex set C , in which C 1 , C 2 ∈ C are joined by an edge if and only if C 1 ∩ C 2 ≠ 0̸ . Erdős conjectured that the chromatic number of triangle-free intersection graphs of n segments in the plane is bounded from above by a constant. Here we show that it is bounded by a polylogarithmic function of n , which is the first nontrivial bound for this problem. More generally, we prove that for any t and k , the chromatic number of every K k -free intersection graph of n curves in the plane, every pair of which have at most t points in common, is at most ( c t log n log k ) c log k , where c is an absolute constant and c t only depends on t . We establish analogous results for intersection graphs of convex sets, x -monotone curves, semialgebraic sets of constant description complexity, and sets that can be obtained as the union of a bounded number of sets homeomorphic to a disk. a mix of results on partially ordered sets and planar separators, for large k we improve the best known upper bound on the number of edges of a k -quasi-planar topological graph with n vertices, that is, a graph drawn in the plane with curvilinear edges, no k of which are pairwise crossing. As another application, we show that for every ε > 0 and for every positive integer t , there exist δ > 0 and a positive integer n 0 such that every topological graph with n ≥ n 0 vertices, at least n 1 + ε edges, and no pair of edges intersecting in more than t points, has at least n δ pairwise intersecting edges.