On the Monotone Upper Bound Problem

Abstract

The Monotone Upper Bound Problem asks for the maximal number {\small $M(d,n)$} of vertices on a strictly increasing edge-path on a simple {\small $d$}-polytope with {\small $n$} facets. More specifically, it asks whether the upper bound

{\small \[ M(d,n)\ \le\ M_{\rm ubt}(d,n) \]}\!\!

provided by McMullen's [McMullen 70] Upper Bound Theorem is tight, where {\small $M_{\rm ubt}(d,n)$} is the number of vertices of a dual-to-cyclic d-polytope with n facets.

It was recently shown that the upper bound {\small $M(d,n)\le M_{\rm ubt}(d,n)$} holds with equality for small dimensions ({\small $d\le 4$} [Pfeifle 04]) and for small corank ({\small $n\le d+2$} [Gärtner et al. 01]). Here we prove that it is not tight in general: in dimension {\small $d=6$}, a polytope with {\small $n=9$} facets can have {\small $ M_{\rm ubt}(6,9)=30$} vertices, but not more than {\small $M(6,9)\le29$} vertices can lie on a strictly increasing edge-path.

The proof involves classification results about neighborly polytopes of small corank, Kalai's [Kalai 88] concept of abstract objective functions, the Holt-Klee conditions [Holt and Klee 98], explicit enumeration, Welzl's extended Gale diagrams [Welzl 01], and randomized generation of instances, as well as nonrealizability proofs via a version of the Farkas lemma.