Unavoidable patterns and plane paths in dense topological graphs

Let Cs,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^k⁵ᵏ, contains a topological subgraph weakly isomorphic to Ck,k. As a corollary, every n-vertex simple topological graph not containing a plane path of length k has at most Oₖ(n2 - 8/k⁴) 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(n4/3) 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.