On Optimal Point Sets Determining Distinct Triangles

Erdős and Fishburn studied the maximum number of points in the plane that span k distances (i. e. the set of pairwise distances between points has cardinality k) and classified these configurations, as an inverse problem of the Erdős 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.