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 subsets of the edge set, 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 statement remains true (1) for topological graphs in which only k-grids consisting of 2k vertex-disjoint edges are forbidden, and (2) for graphs drawn by straight-line edges, with no k-element sets of edges such that every edge in the first set crosses every edge in the other set and each pair of edges within the same set is disjoint. These conjectures are shown to be true apart from log* n and log ² n factors, respectively. We also settle the conjectures for some special cases.