The characterization of diameter-Ramsey simplices by their circumcentres

A simplex is a finite set of affinely independent points in Euclidean space, and it is diameter-Ramsey if every finite coloring of a sufficiently large Euclidean space contains a monochromatic congruent copy of the simplex whose diameter equals the diameter of the original simplex. Its circumcentre is the centre of the unique sphere through its vertices, and its convex hull is the set of all convex combinations of those vertices.

Diameter-Ramsey simplex conjecture. A simplex is diameter-Ramsey if and only if its circumcentre is contained in its convex hull.

This conjecture would characterize diameter-Ramsey simplices geometrically. The paper notes that it would imply that no obtuse triangle is diameter-Ramsey, but does not resolve the question.