Optimal point sets determining few distinct triangles

TL;DR

This paper investigates the maximum size of finite point sets in the plane that determine a fixed number of distinct triangles, providing exact values for small cases and bounds for larger ones.

Contribution

It extends previous work by characterizing the maximum point sets for exactly one and two distinct triangles and establishing bounds for larger numbers.

Findings

F(1) = 4 points

F(2) = 5 points

F(t) < 48(t+1) for all t

Abstract

We generalize work of Erdos and Fishburn to study the structure of finite point sets that determine few distinct triangles. Specifically, we ask for a given $t$, what is the maximum number of points that can be placed in the plane to determine exactly $t$ distinct triangles? Denoting this quantity by $F(t)$, we show that $F(1) = 4$, $F(2) = 5$, and $F(t) < 48(t+1)$ for all $t$. We also completely characterize the optimal configurations for $t = 1, 2$.