Intersection theorems for triangles

TL;DR

This paper investigates the maximum size of intersecting families of triangles formed from a set of points in the plane, establishing an upper bound for points in convex position.

Contribution

It provides a new upper bound on the size of intersecting families of triangles for points in convex position, advancing understanding of geometric intersection properties.

Findings

Maximum intersecting family size is approximately one quarter of all triangles.

For points in convex position, the largest intersecting family contains at most about 25% of all triangles.

The result applies specifically to points arranged in convex position.

Abstract

Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In particular, we show that if a set $P$ of $n$ points is in convex position, then the largest intersecting family of triangles with vertices in $P$ contains at most $(\frac{1}{4}+o(1))\binom{n}{3}$ triangles.