The number of s-dimensional faces in a complex: An analogy between the simplex and the cube

Suppose 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 rs? 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 f...

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Bibliographic Details
Published in:Journal of combinatorial theory 1969, Vol.6 (1), p.86-89
Main Author: Kruskal, J.B.
Format: Article
Language:English
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Summary:Suppose 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 rs? 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 correct. The pertinent formulas for the Harper array are presented.
ISSN:0021-9800
DOI:10.1016/S0021-9800(69)80109-2