On disjoint crossing families in geometric graphs

A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A geometric graph contains a (k,l)-crossing family if there is a pair of edge subsets E 1, E 2 such that | E 1 | = k and | E 2 | = l, the edges in E 1 are pairwise crossing, the edges in E 2 are pairwise crossing, and every edge in E 1 is disjoint to every edge in E 2. We conjecture that for any fixed k, l, every n-vertex geometric graph with no (k, l)-crossing family has at most c k, l n edges, where c k, l is a constant that depends only on k and l. In this note, we show that every n-vertex geometric graph with no (k, k)-crossing family has at most c k nlogn edges, where c k is a constant that depends only on k, by proving a more general result that relates an extremal function of a geometric graph F with an extremal function of two completely disjoint copies of F. We also settle the conjecture for geometric graphs with no (2, 1)-crossing family. As a direct application, this implies that for any circle graph F on three vertices, every n-vertex geometric graph that does not contain a matching whose intersection graph is F has at most O(n) edges.