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Irredundant families of subcubes

Published online by Cambridge University Press:  12 January 2011

DAVID ELLIS
DAVID ELLIS*
Affiliation:
St John's College, Cambridge. e-mail: dce27@cam.ac.uk
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Abstract

We consider the problem of finding the maximum possible size of a family of d-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 [1] conjectured that for d > n/2, the answer is (which is attained by the family of all d-subcubes containing a fixed point). We prove this in the case where all the subcubes go through either (0, 0, . . . , 0) or (1, 1, . . . , 1). We also give a new proof of an upper bound of Meshulam [6]. Finally, we consider the case d ≤ n/2. We give a probabilistic construction showing that in this case, Meshulam's upper bound is correct to within a constant factor. When n = 2d, we construct an irredundant family of size .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 2 , March 2011 , pp. 257 - 272
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

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