A Kruskal-Katona theorem for cubical complexes

The optimal number of faces in cubical complexes which lie in cubes refers to the maximum number of faces that can be constructed from a certain number of faces of lower dimension, or the minimum number of faces necessary to construct a certain number of faces of higher dimension. If m is the number of faces of r in a cubical complex, and if s > r(s < r), then the maximum(minimum) number of faces of dimension s that the complex can have is

m_(s/r) +. (m-m_(r/r))^(s/r), in terms of upper and lower semipowers. The corresponding formula for simplicial complexes, proved independently by J. B. Kruskal and G. A. Katona, is m^(s/r). A proof of the formula for cubical complexes is given in this paper, of which a flawed version appears in a paper by Bernt Lindstrijm. The n-tuples which satisfy the optimaiity conditions for cubical complexes which lie in cubes correspond bijectively with f-vectors of cubical complexes.