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A topological graph G is a graph drawn in the plane with vertices represented by points and edges represented by continuous arcs connecting the vertices. If every edge is drawn as a straightline segment, then G is called a geometric graph. 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 every n-vertex topological graph with no k-grid has O k (n) edges. We conjecturedoi:10.1016/j.comgeo.2014.02.003 fatcat:go2xryyqejcp7l4cjp6sdi54rq