A note on geometric 3-hypergraphs

Abstract

In this note, we prove several Turán-type results on geometric hypergraphs. The two main theorems are (1) every n-vertex geometric 3-hypergraph in the plane with no three strongly crossing edges has at most O(n 2) edges, and (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.