Number of independent subcubes of a hypercube

[twoflower]

Number of "independent" subcubes of a hypercube

Hello, I am trying to solve this problem: I have an n-dimensional hypercube and m of its vertices. Now I want to compute the maximum number of subcubes of the entire hypercube such that:
- each subcube from the set may contain only those m vertices
- no subcube from the set is part of another subcube from the set

Does anybody have any idea?

Thank you very much.


Standa

 

[Eynstone]

I take the 'subcubes' to be hypercubes of lower dimension.
The answer depends on the positions of the m vertices critically. For each (n-1)dimensional face , find the number of vertices on it. If two adjacent faces contain 2 or more vertices, subtract the number of hypercubes on the common 'edge'. Inductively, calculate the number of hypercubes on each (n-r) D face (r=1,2,...).