A structure theorem for pseudosegments and its applications

We prove a far-reaching strengthening of Szemerédi's regularity lemma for intersection graphs of pseudosegments. It shows that the vertex set of such a graph can be partitioned into a bounded number of parts of roughly the same size such that almost all bipartite graphs between different pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every n-vertex simple topological graph with no k pairwise disjoint edges has at most $n{(\log n)}^{O(\log k)}$ edges.