Unavoidable patterns and plane paths in dense topological graphs

TL;DR

This paper investigates unavoidable patterns in dense topological graphs, establishing bounds on edges based on the absence of long plane paths, and introduces new extremal results for specific graph classes.

Contribution

It provides new extremal bounds for simple topological graphs avoiding long plane paths and characterizes the presence of certain bipartite subgraphs in dense configurations.

Findings

Graphs with no long plane paths have subquadratic edge bounds.

For k=3, graphs have at most O(n^{4/3}) edges.

X-monotone graphs without length-3 paths have linear edges.

Abstract

Let $C_{s,t}$ be the complete bipartite geometric graph, with $s$ and $t$ vertices on two distinct parallel lines respectively, and all $s t$ straight-line edges drawn between them. In this paper, we show that every complete bipartite simple topological graph, with parts of size $2(k-1)^4 + 1$ and $2^{k^{5k}}$, contains a topological subgraph weakly isomorphic to $C_{k,k}$. As a corollary, every $n$-vertex simple topological graph not containing a plane path of length $k$ has at most $O_k(n^{2 - 8/k^4})$ edges. When $k = 3$, we obtain a stronger bound by showing that every $n$-vertex simple topological graph not containing a plane path of length 3 has at most $O(n^{4/3})$ edges. We also prove that $x$-monotone simple topological graphs not containing a plane path of length 3 have at most a linear number of edges.