On the Number of Hyperedges in the Hypergraph of Lines and Pseudo-Discs

Consider a hypergraph whose vertex set is a family of n lines in general position in the plane, and whose hyperedges are induced by intersections with a family of pseudo-discs. We prove that the number of t-hyperedges is bounded by O_t(n²) and that the total number of hyperedges is bounded by O(n³). Both bounds are tight.