Unavoidable Patterns in Complete Simple Topological Graphs

We show that every complete n-vertex simple topological graph contains a topological subgraph on at least (log n)(1/4-o(1)) vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound Omega(log(1/8) n) obtained in 2003 by Pach, Solymosi, and Toth. We also show that every complete n-vertex simple topological graph contains a plane path of length at least (log n)(1-o(1)).