Minimum sets forcing monochromatic triangles

The fundamental problem in Euclidean Ramsey theory is the following: Given a configuration C of points in R-n and an arbitrary k-coloring of R-n, does there exist a monochromatic set of points in R-n congruent to C? In this paper we focus on the case where k = n = 2 and C is the vertex set of a triangle. We will say that a triangle T is 2-Ramsey if every 2-coloring of R-2 gives a monochromatic set congruent to the vertex set of T. The foundations of Euclidean Ramsey theory were laid in a sequence of three seminal papers [2], [3], and [4]. Among the many results of these papers, the authors make the following conjecture: