Irredundant Families of Subcubes

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.

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