The classification of finite Euclidean Ramsey sets

OpenAI's classification (2026) of the finite Euclidean Ramsey sets: a finite set is Ramsey exactly when a linear system over the tensor square of its coordinate field has a solution. The colouring obstruction from rational functionals, a finite lattice identity through the flatness of completion, two-scale configurations by Gaussian smoothing, the Hales–Jewett theorem, ultrafilter semigroups and a path criterion in a free group, and the correction of endpoints through derived series. Examples on circles: at most five concyclic points are Ramsey, including kites that embed in no finite transitive set; nine generic points and three rotated squares are not.

Nine lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Ramsey sets and the tensor criterion
2. A colouring obstruction
3. A finite lattice identity
4. Two-scale configurations
5. The Hales–Jewett theorem
6. Ultrafilters and idempotents
7. The path criterion
8. Equal endpoints
9. Examples on circles

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Completion (Commutative algebra for geometry)
• Noetherian and Artinian rings (Commutative algebra for geometry)
• Linear algebra, including the diagonalization of real symmetric matrices, and Zorn's lemma.
• Noetherian rings, the Hilbert basis theorem and the flatness of completion (course Commutative algebra for geometry, lessons Noetherian and Artinian rings and Completion).
• Taylor's theorem, Fubini's theorem, linear changes of variables and differentiation under the integral sign (core courses Real Analysis I and II).