This paper studies results similar to the Ramsey theorem for finite metric spaces. The flavor of the paper may be seen in the definition: \(F_k(\alpha,n)\) is the largest \(m\) such that any \(n\)-point metric space contains an \(m\)-point subspace which is either \(\alpha\)-equivalent to an equilateral space or \(\alpha\)-equivalent to a space for which every triple of points has distortion at least \(k\) from an equilateral triangle.