The classification problem for finite Euclidean Ramsey sets

Theorem 1.1: a finite $A\subset\mathbb R^d$ of at least two points with full affine span is Ramsey iff some $(d+1)\times(d+1)$ matrix $P$ over $F\otimes_{\mathbb Q}F$, with $F$ the field generated by the coordinates, satisfies $(p_i\otimes1)^TP(1\otimes p_i)=0$ for every $p_i=(1,a_i)$ and $m_F(P_{\alpha\beta})=\delta_{\alpha\beta}$ on the spatial block. Consequences: every subtransitive set and every set of at most five points on a circle is Ramsey; nine circle points with algebraically independent parameters, and twelve points forming three rotated squares, are not. The criterion is exact algebra over the coordinate field; the paper says it is not a procedure for deciding the property from numerical coordinates. The Leader-Russell-Walters disproof is a separate entry. Graham's spherical conjecture had already fallen to Palvolgyi's ChatGPT-found seven-point circle set (arXiv:2609.23327), which this manuscript cites; see graham-spherical-ramsey-conjecture.

Theorem 1.1: a finite $A\subset\mathbb R^d$ of at least two points with full affine span is Ramsey iff some $(d+1)\times(d+1)$ matrix $P$ over $F\otimes_{\mathbb Q}F$, with $F$ the field generated by the coordinates, satisfies $(p_i\otimes1)^TP(1\otimes p_i)=0$ for every $p_i=(1,a_i)$ and $m_F(P_{\alpha\beta})=\delta_{\alpha\beta}$ on the spatial block. Consequences: every subtransitive set and every set of at most five points on a circle is Ramsey; nine circle points with algebraically independent parameters, and twelve points forming three rotated squares, are not. The criterion is exact algebra over the coordinate field; the paper says it is not a procedure for deciding the property from numerical coordinates. The Leader-Russell-Walters disproof is a separate entry. Graham's spherical conjecture had already fallen to Palvolgyi's ChatGPT-found seven-point circle set (arXiv:2609.23327), which this manuscript cites; see graham-spherical-ramsey-conjecture.