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Slicing the hypercube is not easy

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arxiv 2102.05536 v2 pith:Y73Q6OT4 submitted 2021-02-10 math.CO cs.AIcs.CC

Gal Yehuda , Amir Yehudayoff This is my paper · ORCID
classification math.COcs.AIcs.CC
keywords hypercubehyperplaneslowerapplicationsboundboundscomplexitycomputational
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abstract

We prove that at least $\Omega(n^{0.51})$ hyperplanes are needed to slice all edges of the $n$-dimensional hypercube. We provide a couple of applications: lower bounds on the computational complexity of parity, and a lower bound on the cover number of the hypercube by skew hyperplanes.

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  1. Improved Upper Bounds for Slicing the Hypercube

    cs.AI 2026-02 conditional novelty 6.0 of 10

    All edges of the n-dimensional hypercube can be sliced with at most 4n/5 hyperplanes (with a small odd-multiple-of-5 exception), improving the 1971 Paterson bound of 5n/6 via an explicit 8-hyperplane slicing of Q10.

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