On Optimal Point Sets Determining Distinct Triangles

TL;DR

This paper investigates the maximum number of points in the plane that determine a limited number of distinct triangles, proving a conjecture about six points spanning three triangles and characterizing the unique optimal configuration.

Contribution

It resolves a conjecture that at most six points can span three distinct triangles and identifies the hexagon as the unique optimal configuration.

Findings

Maximum six points span three triangles

Hexagon is the unique optimal configuration

Optimal sets likely not on the square lattice

Abstract

Erd\H{o}s and Fishburn studied the maximum number of points in the plane that span $k$ distances and classified these configurations, as an inverse problem of the Erd\H{o}s distinct distances problem. We consider the analogous problem for triangles. Past work has obtained the optimal sets for one and two distinct triangles in the plane. In this paper, we resolve a conjecture that at most six points in the plane can span three distinct triangles, and obtain the hexagon as the unique configuration that achieves this. We also provide evidence that optimal sets cannot be on the square lattice in the general case.