Slicing all Edges of an $n$-cube Requires $n^{2/3}$ Hyperplanes

Ohad Klein

Slicing all Edges of an $n$-cube Requires $n^{2/3}$ Hyperplanes

Combinatorics 2022-12-23 v2

Authors: Ohad Klein

Abstract

Consider the $n$ -cube graph with vertices $\{-1,1\}^n$ and edges connecting vertices with hamming distance $1$ . How many hyperplanes in $\mathbb{R}^n$ are needed in order to dissect all edges? We show that at least $\widetilde{\Omega}(n^{2/3})$ are needed, which improves the previous bound of $\Omega(n^{0.51})$ by Yehuda and Yehudayoff.

Keywords

n-cube hyperplanes edge slicing hamming distance lower bound discrete geometry

Cite

@article{arxiv.2212.03328,
  title  = {Slicing all Edges of an $n$-cube Requires $n^{2/3}$ Hyperplanes},
  author = {Ohad Klein},
  journal= {arXiv preprint arXiv:2212.03328},
  year   = {2022}
}

Comments

10 pages; A few inaccuracies corrected

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