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 edges 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 n log n edges, where c k is a constant that depends only on k, by proving a more general result which relates extremal function of a geometric graph F with 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 3 vertices, every n-vertex geometric graph that does not contain a matching whose intersection graph is F has at most O ( n ) edges.