Equilateral Triangles in Finite Metric Spaces

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, . . . , n}, the number Dn 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 D1 = 3, D2 = 6, we prove that D3 = 12, D4 = 33 and 81 ≤ D5 ≤ 95. In classical combinatorial theory the following is a well-known, widely open problem: determine the minimal order of a complete graph such that when coloring the edges with n colors (with n ∈ N fixed) we can find at least one monochromatic triangle. Such a smallest integer has been (among others) proved to exist by Ramsey [10] and is typically denoted by Rn[3, 3, . . . , 3 } {{ }