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.

翻译：设 $C_{s,t}$ 为完全二分几何图，其顶点分别位于两条不同的平行直线上，其中一条有 $s$ 个顶点，另一条有 $t$ 个顶点，且所有 $s t$ 条边均为连接两部分的直线段。本文证明，每个完全二分简单拓扑图，若其两部分的大小分别为 $2(k-1)^4 + 1$ 和 $2^{k^{5k}}$，则必包含一个弱同构于 $C_{k,k}$ 的拓扑子图。作为推论，每个不包含长度为 $k$ 的平面路径的 $n$ 顶点简单拓扑图至多有 $O_k(n^{2 - 8/k^4})$ 条边。当 $k = 3$ 时，我们得到了更强的界：每个不包含长度为 3 的平面路径的 $n$ 顶点简单拓扑图至多有 $O(n^{4/3})$ 条边。我们还证明，不包含长度为 3 的平面路径的 $x$-单调简单拓扑图至多有线性数量的边。