On the Number of Hyperedges in the Hypergraph of Lines and Pseudo-Discs - Keller, Chaya; Keszegh, Balázs; Pálvölgyi, Dömötör (The Electronic Journal of Combinatorics, 2022)
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 $O_t(n^2)$ and that the total number of hyperedges is bounded by $O(n^3)$. Both bounds are tight.