The intersection graph of a collection C of sets is the graph on the vertex set C, in which C-1 . C-2 is an element of C are joined by an edge if and only if C-1 boolean AND C-2 not equal empty set. Erdos 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.