Finite three dimensional partial orders which are not sphere orders

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Abstract. Given a partially ordered set P = (X; P ), a function F which assigns to each x 2 X a set F (x) so that x y in P if and only if F (x) F (y) is called an inclusion representation. Every poset has such a representation, so it is natural to consider restrictions on the nature of the images of the function F . In this paper, we consider inclusion representations assigning to each x 2 X a sphere in Rd, d-dimensional Euclidean space. Posets which have such representations are called sphere orders. When d = 1, a sphere is just an interval from R, and the class of nite posets which have an inclusion representation using intervals from R consists of those posets which have dimension at most two. But when d 2, some posets of arbitrarily large dimension have inclusion representations using spheres in Rd. However, using a theorem of Alon and Scheinerman, we know that not all posets of dimension d + 2 have inclusion representations using spheres in Rd. In 1984, Fishburn and Trotter asked ...

Stefan Felsner, Peter C. Fishburn, William T. Trot

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