Graham's spherical conjecture: every finite spherical set is Ramsey

Claims a seven-point cyclic set, for any transcendental radius parameter $r>2$ (for example $r=\pi$), that is not Ramsey: a finite colouring of $\mathbb R^n$ with no monochromatic congruent copy, in every dimension. Since the set lies on a circle, this disproves the conjecture that every spherical set is Ramsey. The colouring is non-constructive, built from a non-measurable derivation of $\mathbb R$. The unverified appendix adds that almost every seven-point circular set is non-Ramsey and that the method cannot work below seven points. The OpenAI release's classification manuscript (23 September 2026) cites this paper as the disproof, adds a twelve-point spherical example by the same derivation-moment method, and proves every set of at most five circle points Ramsey; six circle points remain open between the two.

Claims a seven-point cyclic set, for any transcendental radius parameter $r>2$ (for example $r=\pi$), that is not Ramsey: a finite colouring of $\mathbb R^n$ with no monochromatic congruent copy, in every dimension. Since the set lies on a circle, this disproves the conjecture that every spherical set is Ramsey. The colouring is non-constructive, built from a non-measurable derivation of $\mathbb R$. The unverified appendix adds that almost every seven-point circular set is non-Ramsey and that the method cannot work below seven points. The OpenAI release's classification manuscript (23 September 2026) cites this paper as the disproof, adds a twelve-point spherical example by the same derivation-moment method, and proves every set of at most five circle points Ramsey; six circle points remain open between the two.