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On the monotone upper bound problem (2003)
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@MISC{Pfeifle03onthe,
author = {Julian Pfeifle and Günter M. Ziegler},
title = {On the monotone upper bound problem},
year = {2003}
}
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Abstract
The Monotone Upper Bound Problem asks for the maximal number M(d, n) ofvertices 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) holdswith 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 =6apolytopewithn =9facetscanhaveMubt(6, 9) = 30 vertices, but not more than 27 ≤ 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.
Keyphrases
monotone upper bound problem strictly-increasing edge-path upper bound holdswith equality neighborly polytopes non-realizability proof small corank dual-to-cyclic d-polytope simple d-polytope gale diagram holt-klee condition abstract objective function rtner et al maximal number classification result upper bound theorem farkas lemma explicit enumeration small dimension
