Unavoidable Patterns in Complete Simple Topological Graphs

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