On grids in topological graphs

A topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. A k-grid in a topological graph is a pair of edge subsets, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that for a fixed constant k, every n-vertex topological graph with no k-grid has O(n) edges. We conjecture that this remains true even when: (1) considering grids with distinct vertices; or (2) all edges are straight-line segments and the edges within each subset of the grid are required to be pairwise disjoint. These conjectures are shown to be true apart from log∗ n and log2 n factors, respectively. We also settle the conjectures for some special cases, including the second conjecture for convex geometric graphs. This result follows from a stronger statement that generalizes the celebrated Marcus-Tardos Theorem on excluded patterns in 0-1 matrices.

• Type: Conference
• Year: 2009
• Where: COMPGEOM
• Authors: Eyal Ackerman, Jacob Fox, János Pach, Andrew Suk
• Keywords: COMPGEOM 2009, Conjecture, Discrete Geometry, N-vertex Topological Graph, Topological Graph