The Boolean lattice B6 is not a sphere order

A partially ordered set (poset) is called a sphere order if it can be represented by spherical balls in a Euclidean space of some dimension, partially ordered by containment. In 1999, Felsner, Fishburn, and Trotter proved that there exist finite posets that are not sphere orders, but their proof is Ramsey-theoretic and does not provide an estimate of the cardinality of such a poset, although they are “tempt[ed] to conjecture that there is a poset of modest size, say at most 100 points, which is not a sphere order”. In this talk I'll show how to prove, using an identification of spherical containment relations with causal orderings of points in Minkowski space, that this conjecture is true: the Boolean lattice of subsets of {1, 2, 3, 4, 5, 6}, ordered by containment, is not a sphere order.