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.
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