
Tanbir Ahmed

Hunter Snevily
Keywords:
Distance sets, Triangular Lattice,
Abstract
A planar pointset $X$ in Euclidean plane is called a $k$distance set if there are exactly $k$ different distances among the points in $X$. The function $g(k)$ denotes the maximum number of points in the Euclidean plane that is a $k$distance set. In 1996, Erdős and Fishburn conjectured that for $k\geq 7$, every $g(k)$point subset of the plane that determines $k$ different distances is similar to a subset of the triangular lattice. We believe that if $g(k)$ is an increasing function of $k$, then the conjecture is false. We present data that supports our claim and a method of construction that unifies known optimal point configurations for $k\geq 3$.