A note on geometric 3-hypergraphs

TL;DR

This paper establishes upper bounds on the number of edges in certain geometric 3-hypergraphs in 2D and 3D, supporting conjectures about their extremal properties.

Contribution

It proves Turán-type bounds for geometric 3-hypergraphs in 2D and 3D, confirming conjectures on their maximum edge counts under specific crossing constraints.

Findings

Geometric 3-hypergraphs in 2D with no three strongly crossing edges have at most O(n^2) edges.

Geometric 3-hypergraphs in 3D with no two disjoint edges have at most O(n^2) edges.

Results support conjectures by Dey and Pach, and by Akiyama and Alon.

Abstract

In this note, we prove several Tur\'an-type results on geometric hypergraphs. The two main theorems are 1) Every $n$-vertex geometric 3-hypergraph in 2-space with no three strongly crossing edges has at most $O(n^2)$ edges, 2) Every $n$-vertex geometric 3-hypergraph in 3-space with no two disjoint edges has at most $O(n^2)$ edges. These results support two conjectures that were raised by Dey and Pach, and by Akiyama and Alon.