Extremal problems for pairs of triangles

TL;DR

This paper determines the maximum sizes of certain convex geometric hypergraphs involving pairs of triangles, extending previous results to non-convex sets and solving open problems about intersecting triangle families.

Contribution

It provides exact extremal functions for seven of eight configurations of pairs of triangles in convex and non-convex point sets, including solving open problems on intersecting triangle families.

Findings

Exact extremal functions for seven configurations of pairs of triangles.

Extension of results to non-convex point sets.

Determination of maximum size of intersecting triangle families.

Abstract

A convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. There are eight pairwise nonisomorphic cgh's consisting of two disjoint triples. These were studied at length by Bra{\ss} (2004) and by Aronov, Dujmovi\'c, Morin, Ooms, and da Silveira (2019). We determine the extremal functions exactly for seven of the eight configurations. The above results are about cyclically ordered hypergraphs. We extend some of them for triangle systems with vertices from a non-convex set. We also solve problems posed by P. Frankl, Holmsen and Kupavskii (2020), in particular, we determine the exact maximum size of an intersecting family of triangles whose vertices come from a set of $n$ points in the plane.