  We consider the problem of finding the maximum possible size of a family of
k-dimensional subcubes of the n-cube {0,1}^{n}, none of which is contained in
the union of the others. (We call such a family `irredundant'). Aharoni and
Holzman conjectured that for k > n/2, the answer is {n choose k} (which is
attained by the family of all k-subcubes containing a fixed point). We give a
new proof of a general upper bound of Meshulam, and we prove that for k >= n/2,
any irredundant family in which all the subcubes go through either (0,0,...,0)
or (1,1,...,1) has size at most {n choose k}. We then give a general lower
bound, showing that Meshulam's upper bound is always tight up to a factor of at
most e.
