子立方体的无冗余族
Irredundant Families of Subcubes
摘要：
我们考虑的问题，找到一个家庭的k-维子立方体的n-立方体{0，1}^{n}的最大可能的大小，其中没有一个是包含在联合的其他。(We称这样的家庭为“无冗余”）。Aharoni和Holzman证明，当k > n/2时，答案是{n选k}（这是由所有包含一个不动点的k-子立方体的族得到的）。本文给出了Meshulam的一个一般上界的新证明，并证明了对k >= n/2，任何子立方体都通过（0，0，.，0)或（1，1，.，1)最大值为{n选k}。然后，我们给出了一个一般下界，表明Meshulam的上界总是紧到一个因素，最多e。
英文摘要：
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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