Algebraic points on a circle plus at most three arbitrary points are Euclidean Ramsey
Theorem B(b): finitely many distinct points on a circle in the Euclidean plane (centre and radius arbitrary reals), all of whose coordinates are algebraic over Q except for the points with indices in a set T of at most three, form a Euclidean Ramsey configuration. Ramsey is OpenAI's definition from result 172 (for every r >= 2 there is D such that every r-colouring of R^D contains a monochromatic congruent copy). The theorem is stated in OpenAI's vocabulary and again in Mathlib vocabulary (Mathlib's Congruent). With the Möbius invariance of the Ramsey property (mobius-ramsey, Theorem C; the…
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