We prove a far-reaching strengthening of Szemer\'edi's regularity lemma for intersection graphs of pseudo-segments. 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.

翻译：我们证明了Szemerédi正则引理在伪线段相交图上的深远加强形式。该结果表明，此类图的顶点集可划分为有界个规模大致相同的子集，使得不同子集对之间几乎所有二部图都是完全图或空图。利用这一结果，我们改进了简单拓扑图中无交边的界，证明每个不含$k$条两两不相交边的$n$顶点简单拓扑图至多具有$n(\log n)^{O(\log k)}$条边。