Finite three dimensional partial orders which are not sphere orders Original Research Article

Given a partially ordered set P = (X, P), a function F which assigns to each x ∈ 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 ∈ 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 finite 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 whether every finite 3-dimensional poset has an inclusion representation using spheres (circles) in R2. In 1989, Brightwell and Winkler asked whether every finite 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 finite 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.