Angle orders and zeros (Q1176975)

An angle order is a partially ordered set whose points can be mapped into unbounded angular regions in the plane such that \(x\) is less than \(y\) in the partial order if and only if \(x\)'s angular region is properly included in \(y\)'s. The zero augmentation of a partially ordered set adds one point to the set that is less than all original points. The authors define special angle orders \(\Gamma_ n\) where the vertices of the angles are related to a fixed circular disk and \(2n\) equidistant points on it. Then they prove that the zero augmentation of \(\Gamma_ n\) is not an angle order when \(n\) is even and sufficiently large. The difficult proof makes extensive use of Ramsey theory. With this result a problem of the authors [ibid. 1, 333-343 (1985; Zbl 0558.06003] is solved.