{"id":"palomar/PALOMAR-2026-10-18-000020","url":"https://github.com/wstrinz/radian-ramsey/tree/37716c59241154354fcc23522253cac7416d6e2c","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-18-000020-v1.json","repo":"https://github.com/wstrinz/radian-ramsey","tags":["math.CO","math.MG","math.NT","msc:05D10","msc:52C10","msc:11J81","euclidean-ramsey","trigonometry","number-theory"],"score":0,"title":"The radian nine-gon is Euclidean Ramsey iff e^(i theta) is algebraic","trust":"high","domain":"math.CO","source":"palomar","authors":["Will Strinz"],"reviews":0,"subject":"math.CO","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-18","reactions":{},"formalized":true}