AbstractSuppose a complex has a fixed number of r-dimensional faces. How many s-dimensional faces can it have? In particular, what is the maximum possible number if r<s, and the minimum possible number if r>s? For a subcomplex of a simplex, this has already been answered in Kruskal [3]. The work of Bernstein [1] and Harper [2] provides an answer for a subcomplex of the cube, if r=0 and s=1. The present paper points out a very strong analogy between the two situations, in which “Harper arrays” in a cube correspond to the “cascade complexes” in a simplex. By analogy, it is plausible to conjecture that simply counting the faces of a Harper array will provide the answer in the cube situation. I believe that this conjecture is almost surely corr...