On the light side of geometric graphs

Let GG be a geometric graph on nn vertices in general position in the plane. Suppose that for every line ℓℓ in the plane the subgraph of GG induced by the set of vertices in one of the two half-planes bounded by ℓℓ has at most kk edges (k≥1k≥1 may be a function of nn). Then GG has at most O(nk) edges. This bound is best possible.