On the Monotone Upper Bound Problem

The Monotone Upper Bound Problem asks for the maximal number M(d, n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d, n) ≤ Mubt(d, n) provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(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 M(d, n) ≤ Mubt(d, n) holds with equality for small dimensions (d ≤ 4: Pfeifle, 2003) and for small corank (n ≤ d+ 2: Gärtner et al., 2001). Here we prove that it is not tight in general: In dimension d = 6 a polytope with n = 9 facets can have Mubt(6, 9) = 30 vertices, but not more than 26 ≤ M(6, 9) ≤ 29 vertices can lie on a strictly-increasing edge-path. The proof involves classification results about neighborly polytopes, Kalai’s (1988) concept of abstract objective functions, the Holt-Klee conditions (1998), explicit enumeration, Welzl’s (2001) extended Gale diagrams, randomized generation of instances, as well as non-realizability proofs via a version of the Farkas lemma. ∗Supported by the European Graduate Program Combinatorics, Geometry, and Computation (GRK 588/2) in Berlin and by the GIF project Combinatorics of Polytopes in Euclidean Spaces (I-624-35.6/1999) ∗∗Partially supported by Deutsche Forschungs-Gemeinschaft (DFG), FZT86, ZI 475/3 and ZI 475/4