On Almost k-Covers of Hypercubes

Abstract

In this paper, we consider the following problem: what is the minimum number of affine hyperplanes in ℝn, such that all the vertices of \(\overrightarrow 0\) are covered at least k times, and \({\left\{{0,1} \right\}^n}\backslash \left\{{\overrightarrow 0} \right\}\) is uncovered? The k = 1 case is the well-known Alon-Füredi theorem which says a minimum of n affine hyperplanes is required, which follows from the Combinatorial Nullstellensatz.

We develop an analogue of the Lubell-Yamamoto-Meshalkin inequality for subset sums, and completely solve the fractional version of this problem, which also provides an asymptotic answer to the integral version for fixed n and k → ∞. We also use a Punctured Combinatorial Nullstellensatz developed by Ball and Serra, to show that a minimum of n + 3 affine hyperplanes is needed for k = 3, and pose a conjecture for arbitrary k and large n.

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Acknowledgement

We would like to thank Noga Alon for sharing the Ramsey-type proof mentioned above, and the anonymous referees for their extremely helpful comments and suggestions on improving the paper.

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Correspondence to Hao Huang.

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Research supported in part by a George W. Woodruff Fellowship.

Research supported in part by the Collaboration Grants from the Simons Foundation.

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Clifton, A., Huang, H. On Almost k-Covers of Hypercubes. Combinatorica 40, 511–526 (2020). https://doi.org/10.1007/s00493-019-4221-y

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Mathematics Subject Classification (2010)

  • 52C17
  • 05B40