The radian nine-gon is Euclidean Ramsey iff e^(i theta) is algebraic
Theorem D: for real theta, the nine points (cos k theta, sin k theta), k = 0, ..., 8, of the Euclidean plane are distinct, lie on the unit circle and are not Euclidean Ramsey when tan(theta/2) is transcendental; and whenever they are distinct, they are Ramsey if and only if cos theta and sin theta are algebraic (that is, iff e^(i theta) is algebraic), if and only if tan(theta/2) is algebraic. This is an iff for a whole one-parameter family, where earlier explicit examples with a transcendental parameter (OpenAI's Proposition 7.7 and Corollary 7.5) settle single sets: the same nine-point…
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