Finite three-dimensional partial orders which are not sphere orders
A partially ordered set \(P\) is called a sphere order if there is some \(d\geq 1\) and a function \(F\) which assigns to each \(p\in P\) a sphere in \(\mathbb{R}^d\) such that \(p\leq q\) in \(P\) iff \(F(p) \subseteqq F(q)\). It is shown that there is an integer \(n_0\) so that if \(n\geq n_0\) the direct product of 3 copies of the chain \(0< 1<\dots< n-1\) is not a sphere order. In the proof the authors use a new extension of the product Ramsey theorem.
scientific article; zbMATH DE number 1334541