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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 remains true even when: (1) considering grids with distinct vertices; or (2) alldoi:10.1145/1542362.1542430 dblp:conf/compgeom/AckermanFPS09 fatcat:lt6ikax4krb5laweowxm7ep4q4