Exact hyperplane covers for subsets of the hypercube

James Aaronson; Carla Groenland; Andrzej Grzesik; Tom Johnston; Bartłomiej Kielak

doi:10.1016/j.disc.2021.112490

Exact hyperplane covers for subsets of the hypercube

Combinatorics 2021-07-02 v4

Authors: James Aaronson , Carla Groenland , Andrzej Grzesik , Tom Johnston , Bartłomiej Kielak

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Abstract

Alon and F\"{u}redi (1993) showed that the number of hyperplanes required to cover $\{0,1\}^n\setminus \{0\}$ without covering $0$ is $n$ . We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering $\{0,1\}^n$ while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left open.

Keywords

incidence geometry of points and lines sphere packings set theory and dimension theory

Cite

@article{arxiv.2010.00315,
  title  = {Exact hyperplane covers for subsets of the hypercube},
  author = {James Aaronson and Carla Groenland and Andrzej Grzesik and Tom Johnston and Bartłomiej Kielak},
  journal= {arXiv preprint arXiv:2010.00315},
  year   = {2021}
}

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Small fixes and updated code

Exact hyperplane covers for subsets of the hypercube

Abstract

Keywords

Cite

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