Unavoidable patterns and plane paths in dense topological graphs

Unavoidable patterns and plane paths in dense topological graphs Bal´azs Keszegh∗ Andrew Suk† G´abor Tardos‡ Ji Zeng§ Abstract Let Cs,t be the complete bipartite geometric graph, with s and t vertices on two distinct parallel lines respectively, and all st 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 2k5k, 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. 1 Introduction A topological graph is a graph drawn in the plane such that its vertices are represented by points and its edges are represented by non-self-intersecting arcs connecting the corresponding points. No edge is allowed to pass through any point representing a vertex other than its endpoints. Tangencies between the edges are not allowed. That is, if two edges share an interior point, then they must properly cross at this point. A topological graph is called plane if there are no crossing edges. Given a topological graph G, we say that H is a topological subgraph of G if V (H) ⊂V (G) and E(H) ⊂E(G). We say that G and H are weakly isomorphic if there is an incidence preserving bijection between the vertices and edges of G and H such that two edges of G cross if and only if the corresponding edges in H cross as well. A topological graph is simple if every pair of its edges intersect at most once: at a common endpoint or at a proper crossing. Simple topological graphs are also known as a simple drawings. If the edges of a topological graph are drawn with straight-line segments, then it is called geometric. We call a geometric graph convex if its vertices are in convex position. In this paper, we are interested in finding large unavoidable patterns in dense simple topological graphs, and in particular, finding large plane paths. Let us emphasize here that a path of length k consists of k + 1 distinct vertices and k edges. It is not hard to see that the simple condition here is necessary for plane paths, as one can easily draw Kn in the plane such that every pair of edges cross. Moreover, a construction due to Pach and T´oth [22] shows that there is a drawing of Kn in the plane such that every pair of edges crosses exactly once or twice. In 2003, Pach, Solymosi, and T´oth [20] showed that every complete n-vertex simple topological graph contains a topological subgraph on Ω(log1/8(n)) vertices that is weakly isomorphic to either a complete convex geometric graph or a so-called complete twisted graph. This bound was later improved by Suk and Zeng [27] to (log n)1/4−o(1). In 1998, Negami proved a bipartite analogue of this theorem. Let Cs,t be a complete bipartite geometric graph with vertex sets U and V , where |U| = s and |V | = t, such that the vertices in U lie on the y-axis and the vertices in V lie on the vertical line x = 1. (It is easy to ∗HUN-REN Alfr´ed R´enyi Institute of Mathematics and ELTE E¨otv¨os Lor´and University, Budapest, Hungary. Supported by the ERC Advanced Grant “ERMiD”, no. 101054936 and by the EXCELLENCE-24 project no. 151504 Combinatorics and Geometry of the NRDI Fund. Email: keszegh@renyi.hu. †Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093 USA. Supported by NSF grant DMS-2246847. Email: asuk@ucsd.edu. ‡HUN-REN Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary. Supported by ERC Advanced Grants “GeoScape”, no. 882971 and “ERMiD”, no. 101054936. Email: tardos@renyi.hu. §HUN-REN Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary. Supported by ERC Advanced Grants “GeoScape”, no. 882971 and “ERMiD”, no. 101054936. Email: zeng.ji@renyi.hu. 1 arXiv:2512.04795v1 [math.CO] 4 Dec 2025 see that Cs,t is determined up to weak isomorphism independent of the exact placement of the vertices on these vertical lines.) In [18], Negami showed that for every k > 1, there is a minimum integer n = n(k) such that every complete bipartite simple topological graph with n vertices in each part contains a topological subgraph weakly isomorphic to Ck,k. The proof in [18] is based on 4-uniform hypergraph Ramsey theory and no explicit bound for n(k) is given. By applying more geometric arguments, we establish the following stronger result. Theorem 1.1. Every complete bipartite simple topological graph with vertex sets U and V , where |U| > 2(k −1)4 and |V | ≥2k5k, contains a topological subgraph weakly isomorphic to Ck,k. We suspect that Theorem 1.1 still holds if |V | is at least single exponential in a power of k, rather than double exponential in Ω(k log k). On the o

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