Exact hyperplane covers for subsets of the hypercube

@article{Aaronson2021ExactHC,
  title={Exact hyperplane covers for subsets of the hypercube},
  author={James Aaronson and C. Groenland and A. Grzesik and Tom Johnston and Bartlomiej Kielak},
  journal={Discret. Math.},
  year={2021},
  volume={344},
  pages={112490}
}
Alon and Furedi (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. 
2 Citations
Constructions in combinatorics via neural networks
We demonstrate how by using a reinforcement learning algorithm, the deep cross-entropy method, one can find explicit constructions and counterexamples to several open conjectures in extremalExpand
Covering Symmetric Sets of the Boolean Cube by Affine Hyperplanes
Alon and Füredi (European J. Combin., 1993) proved that any family of hyperplanes that covers every point of the Boolean cube {0, 1}n except one must contain at least n hyperplanes. We obtain twoExpand

References

SHOWING 1-7 OF 7 REFERENCES
Turán ’ s theorem in the hypercube
We are motivated by the analogue of Turán’s theorem in the hypercube Qn: how many edges can a Qd-free subgraph of Qn have? We study this question through its Ramsey-type variant and obtain asymptoticExpand
On Almost k-Covers of Hypercubes
TLDR
An analogue of the Lubell-Yamamoto-Meshalkin inequality for subset sums is developed, and completely solve the fractional version of this problem, which provides an asymptotic answer to the integral version for fixed $n$ and $k \rightarrow \infty$. Expand
Polynomial Threshold Functions, Hyperplane Arrangements, and Random Tensors
TLDR
The problem of how many low-degree polynomial threshold functions for any higher degrees is settled, showing that $log_2 T(n,d) \approx n \binom{n}{\le d}$. Expand
Domination parameters of hypercubes
  • International Journal of Engineering Science, Advanced Computing and Bio-Technology
  • 2010
Tur[a-acute]n's Theorem in the Hypercube
TLDR
It is shown that for every odd $d$ it is possible to color the edges of Q_n with $\frac{(d+1)^2}{4}$ colors such that each subcube $Q_d$ is polychromatic, that is, contains an edge of each color. Expand
Covering the Cube by Affine Hyperplanes
TLDR
It is proved that any set of hyperplanes that covers all the vertices of the n -cube but one contains at least n hyperplanes is proved. Expand
Exact hyperplane covers for subsets of the hypercube