Unavoidable patterns in complete simple topological graphs

TL;DR

This paper proves that large complete simple topological graphs necessarily contain sizable subgraphs with specific geometric structures and long plane paths, improving previous bounds and revealing unavoidable patterns.

Contribution

It establishes new lower bounds on the size of subgraphs with particular topological properties in complete simple topological graphs, advancing understanding of their structural patterns.

Findings

Existence of large subgraphs isomorphic to convex or twisted graphs

Presence of long plane paths in complete simple topological graphs

Improved bounds over previous results from 2003

Abstract

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 T\'oth. We also show that every complete $n$-vertex simple topological graph contains a plane path of length at least $(\log n)^{1 -o(1)}$.