Equilateral Triangles in Finite Metric Spaces

Abstract

In the context of finite metric spaces with integer distances, we investigate the new Ramsey-type question of how many points can a space contain and yet be free of equilateral triangles. In particular, for finite metric spaces with distances in the set $\{1,\ldots,n\}$, the number $D_n$ is defined as the least number of points the space must contain in order to be sure that there will be an equilateral triangle in it. Several issues related to these numbers are studied, mostly focusing on low values of $n$. Apart from the trivial $D_1=3$, $D_2=6$, we prove that $D_3=12$, $D_4=33$ and $81\leq D_5 \leq 95$.