Neighborly cubical polytopes

Neighborly cubical polytopes exist: for any $n\ge d\ge 2r+2$, there is a cubical convex d-polytope $C_d^n$ whose $r$-skeleton is combinatorially equivalent to that of the $n$-dimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary $\mathrm{\partial }{C}_d^n$ of a neighborly cubical polytope $C_d^n$ maximizes the $f$-vector among all cubical $(d-1)$-spheres with $2^n$ vertices. While we show that this is true for polytopal spheres for $n\le d+1$, we also give a counter-example for $d=4$ and $n=6$. Further, the existence of neighborly cubical polytopes shows that the graph of the $n$-dimensional cube, where $n\ge 5$, is dimensionally ambiguous'' in the sense of Gr\"unbaum. We also show that the graph of the 5-cube is strongly 4-ambiguous’'. In the special case $d=4$, neighborly cubical polytopes have $f_3=f_0/4 \log_2 f_0/4$vertices,sothefacet-vertexratio$f_3/f_0$ is not bounded; this solves a problem of Kalai, Perles and Stanley studied by Jockusch.