Finite Three Dimensional Partial Orderswhich Are Not Sphere

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 R d , 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 R d. However, using a theorem of Alon and Scheinerman, we know that not all posets of dimension d + 2 have inclusion representations using spheres in R d. In 1984, Fishburn and Trotter asked whether every nite 3-dimensional poset has an inclusion representation using spheres (circles) in R 2. In 1989, Brightwell and Winkler asked whether every nite poset is a sphere order and suggested that the answer was negative. In this paper, we settle both questions by showing that there exists a nite 3-dimensional poset which is not a sphere order. The argument requires a new generalization of the Product Ramsey Theorem which we hope will be of independent interest.