Long monotone paths on simple 4-polytopes (Q2382264)

The monotone upper bound problem (Klee, 1965) asks if the maximal number \(M(d,n)\) of vertices in a monotone path along edges of a \(d\)-dimensional polytope with \(n\) facets can be as large as conceivably possible: Is \(M(d,n) = M_{\text{ubt}}(d,n)\), the maximal number of vertices that a \(d\)-polytope with \(n\) facets can have according to the upper bound theorem? We show that in dimension \(d=4\), the answer is ``yes'', despite the fact that it is ``no'' if we restrict ourselves to the dual-to-cyclic polytopes. For each \(n\geq 5\), we exhibit a realization of a polar-to-neighborly 4-dimensional polytope with \(n\) facets and a Hamilton path through its vertices that is monotone with respect to a linear objective function. This constrasts an earlier result, by which no polar-to-neighborly 6-dimensional polytope with 9 facets admits a monotone Hamilton path.