On the number of hyperedges in the hypergraph of lines and pseudo-discs

TL;DR

This paper establishes tight bounds on the number of hyperedges in a hypergraph formed by lines and pseudo-discs in the plane, showing quadratic bounds for fixed hyperedge sizes and cubic bounds overall.

Contribution

It proves tight bounds on the number of hyperedges in hypergraphs of lines and pseudo-discs, advancing understanding of their combinatorial complexity.

Findings

Number of t-hyperedges is O(n^2) for fixed t

Total number of hyperedges is O(n^3)

Bounds are tight and optimal

Abstract

Consider the 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.